CALCU-Magnetism-HW61. An ion with charge q , mass

Solution by Steps

step 1

For question 1, we start by using the formula for the radius of curvature of a charged particle in a magnetic field:

R = \frac{mv}{qB}

step 2

For the second ion with speed

2v

, the radius of curvature

R'

is given by

R' = \frac{m(2v)}{qB} = 2 \cdot \frac{mv}{qB} = 2R

step 3

Therefore, the radius of the second ion's path is

2R

[1] Answer

B

Key Concept

Radius of curvature in a magnetic field

Explanation

The radius of curvature of a charged particle in a magnetic field is directly proportional to its velocity.

Solution by Steps

step 1

For question 2, we use the concept of magnetic force between two parallel currents. According to Ampère's force law, parallel currents attract each other

step 2

Since both wires have currents directed upwards, they will experience a force of attraction

[2] Answer

A

Key Concept

Magnetic force between parallel currents

Explanation

Parallel currents attract each other due to the magnetic fields they generate.

Solution by Steps

step 1

For question 3, we use the right-hand rule for the direction of the magnetic field. The thumb points in the direction of velocity

v

, and the fingers point in the direction of the force (charge separation)

step 2

Since the charge separation is from minus to plus (left to right), the magnetic field must be directed into the page

[3] Answer

A

Key Concept

Right-hand rule for magnetic force

Explanation

The right-hand rule helps determine the direction of the magnetic field based on the direction of velocity and force.

Solution by Steps

step 1

For question 4, we use the fact that a charged particle moving in a magnetic field will experience a force perpendicular to both the velocity and the magnetic field

step 2

Since the proton is undeflected, the electric field must provide a force that cancels the magnetic force

step 3

The magnetic force is directed to the left, so the electric field must be directed to the right to cancel it

[4] Answer

D

Key Concept

Electric and magnetic force balance

Explanation

An electric field can cancel the magnetic force on a charged particle, resulting in no deflection.

Solution by Steps

step 1

For question 5, we use the right-hand rule for the force on a current-carrying wire in a magnetic field

step 2

The current

I

is upwards, and the magnetic field

B

is towards the bottom of the page

step 3

Using the right-hand rule, the force on the left side of the loop is into the page, and the force on the right side is out of the page, causing a clockwise rotation

[5] Answer

A

Key Concept

Right-hand rule for current-carrying wire

Explanation

The right-hand rule helps determine the direction of the force on a current-carrying wire in a magnetic field.

Translate two Chinese

I'm sorry, I can only help with math questions. If you have any math-related queries, feel free to ask!

Generated Graph

Solution by Steps

step 1

To expand the graph of

f(x) = x^2

vertically and reflect it over the

x

-axis, we need to multiply

f(x)

by a negative constant greater than 1

step 2

The function

f(x) = -3x^2

achieves this transformation

12 Answer

G

Key Concept

Vertical Stretch and Reflection

Explanation

Multiplying a function by a negative constant greater than 1 reflects it over the

x

-axis and stretches it vertically.

Solution by Steps

step 1

The parent function is

p(x) = x^3

step 2

The function

p(x) = (x-7)^3

translates the graph to the right by 7 units

13 Answer

D

Key Concept

Horizontal Translation

Explanation

Subtracting a constant from

x

in the function

p(x)

translates the graph to the right by that constant.

Solution by Steps

step 1

The function

f(x) = |x^3|

represents the absolute value of

x^3

step 2

The graph of

|x^3|

is the same as

x^3

for

x \geq 0

and the reflection of

x^3

over the

x

-axis for x < 0

14 Answer

F

Key Concept

Absolute Value Function

Explanation

The absolute value function reflects the negative part of the graph over the

x

-axis.

Solution by Steps

step 1

Given

f(x) = x - 3

and

g(x) = 2x - 4

, we need to find

(f + g)(x)

step 2

(f + g)(x) = (x - 3) + (2x - 4) = 3x - 7

15 Answer

A

Key Concept

Function Addition

Explanation

To add two functions, add their corresponding expressions.

Solution by Steps

step 1

The coupon function is

c(x) = 0.8x

(20% off means paying 80%)

step 2

The delivery function is

d(x) = x + 3

step 3

The composition function for the total amount paid is

d(c(x)) = 0.8x + 3

16 Answer

J

Key Concept

Composition of Functions

Explanation

The composition of functions applies one function to the result of another.

Solution by Steps

step 1

Given

f(x) = x^2 + 1

and

g(x) = 2x

, we need to find

[f \circ g](x)

step 2

[f \circ g](x) = f(g(x)) = f(2x) = (2x)^2 + 1 = 4x^2 + 1

17 Answer

D

Key Concept

Function Composition

Explanation

To compose two functions, substitute the inner function into the outer function.

Solution by Steps

step 1

To find the inverse of

f(x) = 2x + 9

, we swap

x

and

y

and solve for

y

step 2

x = 2y + 9 \implies y = \frac{x - 9}{2}

18 Answer

H

Key Concept

Inverse Function

Explanation

To find the inverse of a function, solve for

x

in terms of

y

and then swap

x

and

y

.

Solution by Steps

step 1

The function

f(x) = x^3

has an inverse that is also a function because it is one-to-one

19 Answer

D

Key Concept

One-to-One Function

Explanation

A function has an inverse that is also a function if it is one-to-one.

Solution by Steps

step 1

Henri's plant grows 3 inches per year starting at 18 inches

step 2

The function representing the height of the plant after

x

years is

f(x) = 3x + 18

20 Answer

J

Key Concept

Linear Growth Function

Explanation

A linear growth function represents a constant rate of change over time.

Solution by Steps

step 1

Given

f(x) = \sqrt{x + 2}

and

(g \circ f)(x) = x - 1

, we need to find

g(x)

step 2

Let

u = f(x) = \sqrt{x + 2}

. Then

g(u) = u^2 - 1

step 3

Therefore,

g(x) = x^2 - 1

Bonus Answer

g(x) = x^2 - 1

Key Concept

Composition and Inverse Functions

Explanation

To find

g(x)

, express

g

in terms of

f

and solve for

g

.

Generated Graph

Solution by Steps

step 1

To solve

1 - \sqrt{2 - \sqrt{3 - x}} \geq 0

for

x

, we start by isolating the square root term

step 2

1 \geq \sqrt{2 - \sqrt{3 - x}}

step 3

Squaring both sides, we get

1 \geq 2 - \sqrt{3 - x}

step 4

Rearranging, we have

\sqrt{3 - x} \geq 1

step 5

Squaring both sides again, we get

3 - x \geq 1

step 6

Solving for

x

, we find

x \leq 2

step 7

Additionally, since

\sqrt{3 - x}

must be real,

3 - x \geq 0

or

x \leq 3

step 8

Combining these, we get

-1 \leq x \leq 2

Question 1 Answer

-1 <= x <= 2

Key Concept

Solving inequalities involving nested square roots

Explanation

Isolate the square root terms and square both sides to eliminate the square roots, then solve the resulting inequalities.

Solution by Steps

step 1

To find the absolute maximum of

f(x) = \frac{1}{1 + |x|} + \frac{1}{1 + |x - 2|}

, we need to evaluate the function at critical points and endpoints

step 2

The function is piecewise, so we consider

x \geq 0

and x < 0 separately

step 3

For

x \geq 0

,

f(x) = \frac{1}{1 + x} + \frac{1}{1 + |x - 2|}

step 4

For x < 0,

f(x) = \frac{1}{1 - x} + \frac{1}{1 + |x - 2|}

step 5

Evaluating at

x = 0

,

f(0) = \frac{1}{1 + 0} + \frac{1}{1 + 2} = 1 + \frac{1}{3} = \frac{4}{3}

step 6

Evaluating at

x = 2

,

f(2) = \frac{1}{1 + 2} + \frac{1}{1 + 0} = \frac{1}{3} + 1 = \frac{4}{3}

step 7

Evaluating at

x = -2

,

f(-2) = \frac{1}{1 - (-2)} + \frac{1}{1 + | -2 - 2|} = \frac{1}{3} + \frac{1}{5} = \frac{8}{15}

step 8

The maximum value is

\frac{4}{3}

at

x = 0

and

x = 2

Question 2 Answer

4/3 at x = 0

Key Concept

Finding absolute maximum of piecewise functions

Explanation

Evaluate the function at critical points and endpoints to determine the maximum value.

Solution by Steps

step 1

To find the radius of the inscribed circle in a right triangle with sides

a

,

b

, and

c

, we use the formula for the inradius

r

step 2

The formula for the inradius

r

is

r = \frac{a + b - c}{2}

step 3

For a right triangle,

c

is the hypotenuse, and

a

and

b

are the legs

step 4

The area of the triangle is

\frac{1}{2}ab

step 5

The semi-perimeter

s

is

\frac{a + b + c}{2}

step 6

The inradius

r

can also be found using

r = \frac{A}{s}

, where

A

is the area

step 7

Substituting the values,

r = \frac{\frac{1}{2}ab}{\frac{a + b + c}{2}} = \frac{ab}{a + b + c}

Question 3 Answer

\frac{ab}{a + b + c}

Key Concept

Inradius of a right triangle

Explanation

The inradius can be found using the area and semi-perimeter of the triangle.

Solution by Steps

step 1

To determine which relation is a function, we need to check if each input has exactly one output

step 2

A function is defined such that for every

x

in the domain, there is exactly one

y

in the range

step 3

Option A:

x y = 2

is not a function because

x

can have multiple

y

values

step 4

Option B:

x = y^2

is not a function because

y

can have multiple

x

values

step 5

Option C:

y = x^2

is a function because each

x

has exactly one

y

step 6

Option D:

y = 3

is a function because each

x

has exactly one

y

Question 4 Answer

C

Key Concept

Definition of a function

Explanation

A function has exactly one output for each input.

Solution by Steps

step 1

To find

f(4)

for

f(x) = x^2 - 2x

, we substitute

x = 4

into the function

step 2

f(4) = 4^2 - 2 \cdot 4

step 3

f(4) = 16 - 8

step 4

f(4) = 8

Question 5 Answer

H

Key Concept

Evaluating a function

Explanation

Substitute the given value into the function and simplify.

Solution by Steps

step 1

To find the zero of

f(x) = -\frac{2}{3}x - 12

, we set

f(x) = 0

step 2

0 = -\frac{2}{3}x - 12

step 3

Solving for

x

, we get

-\frac{2}{3}x = 12

step 4

x = -18

Question 6 Answer

A

Key Concept

Finding the zero of a function

Explanation

Set the function equal to zero and solve for the variable.

Solution by Steps

step 1

To find

f(-2)

for

f(x) = \left\{ \begin{array}{r} |4x| \text{ if } x &lt; -2 \\ x^3 - 1 \text{ if } x \geq -2 \end{array} \right.

, we determine which piece of the function to use

step 2

Since

x = -2

, we use the second piece:

x^3 - 1

step 3

f(-2) = (-2)^3 - 1

step 4

f(-2) = -8 - 1

step 5

f(-2) = -9

Question 7 Answer

J

Key Concept

Evaluating a piecewise function

Explanation

Determine which piece of the function to use based on the given value.

Solution by Steps

step 1

To determine which relation is symmetric with respect to the

x

-axis, we check if replacing

y

with

-y

results in the same equation

step 2

Option A:

xy = 2

is not symmetric because

x(-y) = -2

step 3

Option B:

x = y^2

is not symmetric because

x = (-y)^2 = y^2

step 4

Option C:

y = x^2

is not symmetric because

-y = x^2

step 5

Option D:

y = 3

is symmetric because

-y = -3

is the same as

y = 3

Question 8 Answer

D

Key Concept

Symmetry with respect to the

x

-axis

Explanation

A relation is symmetric with respect to the

x

-axis if replacing

y

with

-y

results in the same equation.

Solution by Steps

step 1

To determine which function is an odd function, we check if

f(-x) = -f(x)

step 2

Option F:

f(x) = -x^3 + 4

is not odd because

f(-x) = -(-x)^3 + 4 = x^3 + 4 \neq -f(x)

step 3

Option G:

f(x) = 2x^3

is odd because

f(-x) = 2(-x)^3 = -2x^3 = -f(x)

step 4

Option H:

f(x) = x^4 - 9

is not odd because

f(-x) = (-x)^4 - 9 = x^4 - 9 \neq -f(x)

step 5

Option J:

f(x) = x^4 + 4x

is not odd because

f(-x) = (-x)^4 + 4(-x) = x^4 - 4x \neq -f(x)

Question 9 Answer

G

Key Concept

Odd function

Explanation

A function is odd if

f(-x) = -f(x)

.

Solution by Steps

step 1

To determine which function has a removable discontinuity, we look for a factor that can be canceled

step 2

Option A:

f(x) = \frac{x}{x+3}

has a discontinuity at

x = -3

, but it is not removable

step 3

Option B:

f(x) = \frac{x^2 - 4}{x + 2}

has a removable discontinuity at

x = -2

because

x^2 - 4 = (x + 2)(x - 2)

step 4

Option C:

f(x) = \frac{1}{x + 3}

has a discontinuity at

x = -3

, but it is not removable

step 5

Option D:

f(x) = x^3 - 3

has no discontinuities

Question 10 Answer

B

Key Concept

Removable discontinuity

Explanation

A removable discontinuity occurs when a factor can be canceled from the numerator and denominator.

Solution by Steps

step 1

To find

\lim_{x \to \infty} \frac{3}{x - 4}

, we analyze the behavior of the function as

x

approaches infinity

step 2

As

x \to \infty

, the term

x - 4

also approaches infinity

step 3

Therefore,

\frac{3}{x - 4} \to 0

Question 11 Answer

G

Key Concept

Limit as

x

approaches infinity

Explanation

As

x

approaches infinity, the denominator grows without bound, causing the fraction to approach zero.

Solution by Steps

step 1

To find the relative maximum of

p(x) = -5x^3 + 47x^2 - 109x + 90

, we need to find the critical points by taking the derivative and setting it to zero

step 2

p'(x) = -15x^2 + 94x - 109

step 3

Setting

p'(x) = 0

, we solve

-15x^2 + 94x - 109 = 0

step 4

Using the quadratic formula,

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

, we find the critical points

step 5

x = \frac{-94 \pm \sqrt{94^2 - 4(-15)(-109)}}{2(-15)}

step 6

Simplifying, we find

x \approx 1.5

and

x \approx 4.5

step 7

Evaluating

p(x)

at these points, we find the relative maximum at

x \approx 4.5

Question 12 Answer

97

Key Concept

Finding relative maximum

Explanation

Take the derivative, set it to zero, and solve for critical points. Evaluate the function at these points to find the maximum.

Solution by Steps

step 1

To find the average rate of change of

h(t) = -16t^2 + 72t

from

t = 3

to

t = 4

, we use the formula

\frac{h(4) - h(3)}{4 - 3}

step 2

h(4) = -16(4)^2 + 72(4) = -256 + 288 = 32

step 3

h(3) = -16(3)^2 + 72(3) = -144 + 216 = 72

step 4

The average rate of change is

\frac{32 - 72}{4 - 3} = \frac{-40}{1} = -40 \text{ ft/s}

Question 13 Answer

G

Key Concept

Average rate of change

Explanation

Use the difference quotient

\frac{f(b) - f(a)}{b - a}

to find the average rate of change over an interval.

Solution by Steps

step 1

To find

f(-4.5)

for

f(x) = \lfloor x - 1 \rfloor

, we substitute

x = -4.5

into the function

step 2

f(-4.5) = \lfloor -4.5 - 1 \rfloor

step 3

f(-4.5) = \lfloor -5.5 \rfloor

step 4

f(-4.5) = -6

Question 14 Answer

A

Key Concept

Floor function

Explanation

The floor function

\lfloor x \rfloor

returns the greatest integer less than or equal to

x

.

}

Solution by Steps

step 1

Identify the points through which the first line passes:

(-4, 1)

,

(-3, 2)

, and

(-2, 3)

step 2

Since the line extends infinitely in both directions, the domain is all real numbers:

(-\infty, \infty)

step 3

The range of the first line is also all real numbers:

(-\infty, \infty)

step 4

Identify the points through which the second line passes:

(-1, -1)

,

(-2, 0)

, and

(-3, 1)

step 5

The second line is a segment, so its domain is

[-3, -1]

step 6

The range of the second line is

[-1, 1]

Answer

Domain:

(-\infty, \infty)

for the first line,

[-3, -1]

for the second line. Range:

(-\infty, \infty)

for the first line,

[-1, 1]

for the second line.

Key Concept

Domain and Range

Explanation

The domain and range of a function describe the set of possible input values (x-values) and output values (y-values), respectively.

Question 2: Given

f(x)=2x^2-x+10

, find

f(4a-2)

.

step 1

Substitute

x = 4a - 2

into the function:

f(4a-2) = 2(4a-2)^2 - (4a-2) + 10

step 2

Expand

(4a-2)^2

:

(4a-2)^2 = 16a^2 - 16a + 4

step 3

Substitute back into the function:

f(4a-2) = 2(16a^2 - 16a + 4) - (4a-2) + 10

step 4

Simplify:

f(4a-2) = 32a^2 - 32a + 8 - 4a + 2 + 10

step 5

Combine like terms:

f(4a-2) = 32a^2 - 36a + 20

Answer

f(4a-2) = 32a^2 - 36a + 20

Key Concept

Function Substitution

Explanation

Substituting a variable expression into a function involves replacing the variable with the given expression and simplifying.

Question 3: Find the zeros of

f(x)=5x^2+32x-21

.

step 1

Use the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where

a = 5

,

b = 32

, and

c = -21

step 2

Calculate the discriminant:

b^2 - 4ac = 32^2 - 4(5)(-21) = 1024 + 420 = 1444

step 3

Find the square root of the discriminant:

\sqrt{1444} = 38

step 4

Substitute into the quadratic formula:

x = \frac{-32 \pm 38}{10}

step 5

Solve for

x

:

x = \frac{-32 + 38}{10} = \frac{6}{10} = 0.6

and

x = \frac{-32 - 38}{10} = \frac{-70}{10} = -7

Answer

The zeros are

x = 0.6

and

x = -7

.

Key Concept

Quadratic Formula

Explanation

The quadratic formula is used to find the roots of a quadratic equation by solving for

x

.

Question 4: Find

f(-10.1)

for

f(x)=\left\{\begin{array}{r}-|2x-9| \text{ if } x&lt;-4 \\ 3.5x^3 \text{ if } x \geq -4\end{array}\right.

.

step 1

Determine which piece of the piecewise function to use: Since -10.1 < -4, use

f(x) = -|2x-9|

step 2

Substitute

x = -10.1

into the function:

f(-10.1) = -|2(-10.1)-9|

step 3

Simplify inside the absolute value:

2(-10.1) = -20.2

, so

-20.2 - 9 = -29.2

step 4

Take the absolute value:

|-29.2| = 29.2

step 5

Apply the negative sign:

f(-10.1) = -29.2

Answer

f(-10.1) = -29.2

Key Concept

Piecewise Functions

Explanation

Piecewise functions are defined by different expressions depending on the value of the input variable.

Question 5: Determine whether the graph of

xy=5

is symmetric with respect to the

x

-axis, the

y

-axis, or the origin.

step 1

Test for symmetry with respect to the

x

-axis: Replace

y

with

-y

in the equation

xy = 5

step 2

The equation becomes

x(-y) = 5 \Rightarrow -xy = 5

, which is not equivalent to the original equation

step 3

Test for symmetry with respect to the

y

-axis: Replace

x

with

-x

in the equation

xy = 5

step 4

The equation becomes

(-x)y = 5 \Rightarrow -xy = 5

, which is not equivalent to the original equation

step 5

Test for symmetry with respect to the origin: Replace

x

with

-x

and

y

with

-y

in the equation

xy = 5

step 6

The equation becomes

(-x)(-y) = 5 \Rightarrow xy = 5

, which is equivalent to the original equation

Answer

The graph is symmetric with respect to the origin.

Key Concept

Symmetry in Graphs

Explanation

A graph is symmetric with respect to the origin if replacing

x

with

-x

and

y

with

-y

yields an equivalent equation.

Question 6: Determine whether the function

f(x)=2(x-1)^2+4x

is even, odd, or neither.

step 1

Test for evenness: Replace

x

with

-x

in the function

f(x)

step 2

f(-x) = 2(-x-1)^2 + 4(-x)

step 3

Simplify:

f(-x) = 2(x^2 + 2x + 1) - 4x = 2x^2 + 4x + 2 - 4x = 2x^2 + 2

step 4

Compare

f(-x)

with

f(x)

:

f(x) = 2(x-1)^2 + 4x = 2x^2 - 4x + 2 + 4x = 2x^2 + 2

step 5

Since

f(-x) = f(x)

, the function is even

Answer

The function is even.

Key Concept

Even and Odd Functions

Explanation

A function is even if

f(-x) = f(x)

for all

x

in the domain.

Question 7: Determine whether

f(x)=\frac{x^2-3x}{x^2-x-6}

is continuous at

x=3

. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

step 1

Factor the denominator:

x^2 - x - 6 = (x-3)(x+2)

step 2

Rewrite the function:

f(x) = \frac{x^2 - 3x}{(x-3)(x+2)}

step 3

Simplify the numerator:

x^2 - 3x = x(x-3)

step 4

Cancel common factors:

f(x) = \frac{x(x-3)}{(x-3)(x+2)} = \frac{x}{x+2}

for

x \neq 3

step 5

Since

x = 3

makes the denominator zero,

f(x)

is undefined at

x = 3

step 6

The discontinuity at

x = 3

is removable because the limit exists but the function is not defined at that point

Answer

The function is discontinuous at

x = 3

with a removable discontinuity.

Key Concept

Removable Discontinuity

Explanation

A removable discontinuity occurs when a function is not defined at a point, but the limit exists at that point.

Question 8: Describe the end behavior of

g(x)=\frac{-4x+1}{x+2}

.

step 1

Determine the horizontal asymptote by comparing the degrees of the numerator and denominator

step 2

Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients:

\frac{-4}{1} = -4

step 3

As

x \to \infty

,

g(x) \to -4

step 4

As

x \to -\infty

,

g(x) \to -4

Answer

The end behavior is

g(x) \to -4

as

x \to \pm \infty

.

Key Concept

End Behavior of Rational Functions

Explanation

The end behavior of a rational function is determined by the degrees of the numerator and denominator and their leading coefficients.

Question 9: When should Mrs. Custer sell the soybeans in order to maximize her income? What will this income be?

step 1

Let

t

be the number of weeks Mrs. Custer waits to sell

step 2

The price per bushel after

t

weeks is

6 + 0.10t

step 3

The number of bushels after

t

weeks is

100 - t

step 4

The income function is

I(t) = (6 + 0.10t)(100 - t)

step 5

Expand the income function:

I(t) = 600 + 10t - 0.10t^2

step 6

To find the maximum income, take the derivative and set it to zero:

I'(t) = 10 - 0.20t = 0

step 7

Solve for

t

:

t = 50

step 8

Substitute

t = 50

back into the income function:

I(50) = (6 + 0.10(50))(100 - 50) = 11 \times 50 = 550

Answer

Mrs. Custer should sell the soybeans after 50 weeks to maximize her income, which will be $550.

Key Concept

Optimization

Explanation

Optimization involves finding the maximum or minimum value of a function, often by taking the derivative and setting it to zero.

Question 10: Find the average speed of the firework from 3.5 to 5.5 seconds.

step 1

The height function is

h(t) = -16t^2 + 106t + 8.5

step 2

Find

h(3.5)

:

h(3.5) = -16(3.5)^2 + 106(3.5) + 8.5 = -196 + 371 + 8.5 = 183.5

step 3

Find

h(5.5)

:

h(5.5) = -16(5.5)^2 + 106(5.5) + 8.5 = -484 + 583 + 8.5 = 107.5

step 4

The change in height is

h(5.5) - h(3.5) = 107.5 - 183.5 = -76

step 5

The time interval is

5.5 - 3.5 = 2

seconds

step 6

The average speed is

\frac{\text{change in height}}{\text{time interval}} = \frac{-76}{2} = -38

feet per second

Answer

The average speed is

-38

feet per second.

Key Concept

Average Speed

Explanation

Average speed is calculated by dividing the total change in position by the total time taken.

Question 11: Given

f(x)=-1.5 \llbracket 0.75x-1.9 \rrbracket

, what is

f(-14)

?

step 1

Substitute

x = -14

into the function:

f(-14) = -1.5 \llbracket 0.75(-14) - 1.9 \rrbracket

step 2

Simplify inside the brackets:

0.75(-14) = -10.5

, so

-10.5 - 1.9 = -12.4

step 3

The floor function

\llbracket -12.4 \rrbracket

is

-13

step 4

Multiply by

-1.5

:

f(-14) = -1.5 \times -13 = 19.5

Answer

f(-14) = 19.5

Key Concept

Floor Function

Explanation

The floor function

\llbracket x \rrbracket

returns the greatest integer less than or equal to

x

.

}

Generated Graph

Solution by Steps

Use transformations of the parent graph of $m(x)=\llbracket x \rrbracket$ to sketch the graph of $p(x)=2 \llbracket x-3 \rrbracket$.

step 1

The parent function is

m(x)=\llbracket x \rrbracket

, which is the floor function

step 2

The transformation

x-3

inside the floor function shifts the graph 3 units to the right

step 3

The multiplication by 2 outside the floor function scales the graph vertically by a factor of 2

step 4

Therefore,

p(x)=2 \llbracket x-3 \rrbracket

is the graph of

m(x)=\llbracket x \rrbracket

shifted 3 units to the right and stretched vertically by a factor of 2

Answer

The graph of

p(x)=2 \llbracket x-3 \rrbracket

is the graph of

m(x)=\llbracket x \rrbracket

shifted 3 units to the right and stretched vertically by a factor of 2.

Key Concept

Transformations of the floor function

Explanation

Shifting and scaling transformations applied to the floor function result in the graph of

p(x)

.

Question 13 Describe the transformations relating the graph of $g(x)=\frac{1}{2}|-6 x-3|+4$ to the graph of its parent function $f(x)=|x|$.

step 1

The parent function is

f(x)=|x|

step 2

The transformation

-6x

inside the absolute value reflects the graph across the y-axis and scales it horizontally by a factor of

\frac{1}{6}

step 3

The transformation

-3

inside the absolute value shifts the graph 3 units to the left

step 4

The multiplication by

\frac{1}{2}

outside the absolute value scales the graph vertically by a factor of

\frac{1}{2}

step 5

The addition of 4 outside the absolute value shifts the graph 4 units up

Answer

The graph of

g(x)=\frac{1}{2}|-6 x-3|+4

is the graph of

f(x)=|x|

reflected across the y-axis, scaled horizontally by

\frac{1}{6}

, shifted 3 units to the left, scaled vertically by

\frac{1}{2}

, and shifted 4 units up.

Key Concept

Transformations of the absolute value function

Explanation

Reflections, scalings, and shifts applied to the absolute value function result in the graph of

g(x)

.

Question 14 Graph $f(x)=\frac{1}{|x|}-1$.

step 1

The parent function is

f(x)=\frac{1}{|x|}

step 2

The transformation

-1

outside the function shifts the graph 1 unit down

step 3

The graph of

f(x)=\frac{1}{|x|}-1

has vertical asymptotes at

x=0

and horizontal asymptote at

y=-1

step 4

The graph is symmetric with respect to the y-axis

Answer

The graph of

f(x)=\frac{1}{|x|}-1

has vertical asymptotes at

x=0

, a horizontal asymptote at

y=-1

, and is symmetric with respect to the y-axis.

Key Concept

Graphing rational functions with absolute values

Explanation

Shifting the graph of

\frac{1}{|x|}

down by 1 unit results in the graph of

f(x)

.

Question 15 If $f(x)=2 x-1$ and $g(x)=\frac{1}{2 x^{2}}$, find $\left(\frac{f}{g}\right)(x)$ and its domain.

step 1

The functions are

f(x)=2 x-1

and

g(x)=\frac{1}{2 x^{2}}

step 2

The quotient function is

\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{2 x-1}{\frac{1}{2 x^{2}}}

step 3

Simplify the quotient:

\left(\frac{f}{g}\right)(x)=(2 x-1) \cdot (2 x^{2})=4 x^{3}-2 x^{2}

step 4

The domain of

\left(\frac{f}{g}\right)(x)

is all real numbers except

x=0

(where

g(x)

is undefined)

Answer

\left(\frac{f}{g}\right)(x)=4 x^{3}-2 x^{2}

with domain

x \neq 0

.

Key Concept

Quotient of functions

Explanation

The quotient of two functions is found by dividing the functions and simplifying, with the domain excluding points where the denominator is zero.

Question 16 A cone with a fixed height of 81 millimeters is shown on a computer screen. An animator increases the radius $r$ at a rate of 4.5 centimeters per minute. Write the function that gives the volume $v(r)$ of the cone in cubic centimeters as a function of time $f(t)$. Assume the radius is 4.5 centimeters at $t=1$.

step 1

The volume of a cone is given by

V=\frac{1}{3}\pi r^{2}h

step 2

Given

h=81

mm = 8.1 cm, the volume function is

V(r)=\frac{1}{3}\pi r^{2}(8.1)=2.7\pi r^{2}

step 3

The radius

r

increases at a rate of 4.5 cm/min, so

r(t)=4.5t

step 4

Substitute

r(t)

into

V(r)

:

V(t)=2.7\pi (4.5t)^{2}=2.7\pi (20.25t^{2})=54.675\pi t^{2}

Answer

The volume function is

V(t)=54.675\pi t^{2}

cubic centimeters.

Key Concept

Volume of a cone as a function of time

Explanation

The volume of a cone can be expressed as a function of time by substituting the time-dependent radius into the volume formula.

Question 17 If $f(x)=3 x^{2}+4$ and $g(x)=\frac{1}{x^{2}-x}$, find $[g \circ f](x)$.

step 1

The functions are

f(x)=3 x^{2}+4

and

g(x)=\frac{1}{x^{2}-x}

step 2

The composition

[g \circ f](x)=g(f(x))=g(3 x^{2}+4)

step 3

Substitute

f(x)

into

g(x)

:

g(3 x^{2}+4)=\frac{1}{(3 x^{2}+4)^{2}-(3 x^{2}+4)}

step 4

Simplify the expression:

g(3 x^{2}+4)=\frac{1}{9 x^{4}+24 x^{2}+16-3 x^{2}-4}=\frac{1}{9 x^{4}+21 x^{2}+12}

Answer

[g \circ f](x)=\frac{1}{9 x^{4}+21 x^{2}+12}

.

Key Concept

Composition of functions

Explanation

The composition of two functions involves substituting one function into the other and simplifying the result.

Question 18 Find the inverse of $f(x)=\frac{3}{x-2}$.

step 1

The function is

f(x)=\frac{3}{x-2}

step 2

To find the inverse, swap

x

and

y

:

x=\frac{3}{y-2}

step 3

Solve for

y

:

x(y-2)=3 \Rightarrow y-2=\frac{3}{x} \Rightarrow y=\frac{3}{x}+2

step 4

The inverse function is

f^{-1}(x)=\frac{3}{x}+2

Answer

The inverse function is

f^{-1}(x)=\frac{3}{x}+2

.

Key Concept

Finding the inverse of a function

Explanation

To find the inverse of a function, swap the variables and solve for the new dependent variable.

Question 19 Determine if $f(x)=\frac{1}{2 x^{2}}$ is a one-to-one function.

step 1

The function is

f(x)=\frac{1}{2 x^{2}}

step 2

A function is one-to-one if each

y

value corresponds to exactly one

x

value

step 3

f(x)=\frac{1}{2 x^{2}}

is not one-to-one because

f(x)=f(-x)

step 4

Therefore,

f(x)=\frac{1}{2 x^{2}}

is not injective (one-to-one)

Answer

f(x)=\frac{1}{2 x^{2}}

is not a one-to-one function.

Key Concept

One-to-one functions

Explanation

A function is one-to-one if each output is associated with a unique input.

Question 20 A construction worker orders 50 boxes of screws. Some are wood screws at \$2.93 per box and some are sheet metal screws at \$4.07 per box. Write the function that can be used to find the number of boxes of wood screws ordered if given the total price paid.

step 1

Let

x

be the number of boxes of wood screws

step 2

The number of boxes of sheet metal screws is

50-x

step 3

The total price is given by

2.93x + 4.07(50-x)

step 4

Simplify the expression:

2.93x + 4.07(50-x) = 2.93x + 203.5 - 4.07x = 203.5 - 1.14x

step 5

The function for the total price is

P(x)=203.5 - 1.14x

Answer

The function for the total price is

P(x)=203.5 - 1.14x

.

Key Concept

Linear functions in cost problems

Explanation

The total cost can be expressed as a linear function of the number of boxes of wood screws.

Bonus Question Given the graph of $p(x)$, sketch the graph of $y=-2 p\left[\frac{1}{2}(x-2)\right]+2$.

step 1

The given transformation involves

p(x)

step 2

The transformation

\frac{1}{2}(x-2)

inside the function scales the graph horizontally by a factor of 2 and shifts it 2 units to the right

step 3

The multiplication by

-2

outside the function reflects the graph across the x-axis and scales it vertically by a factor of 2

step 4

The addition of 2 outside the function shifts the graph 2 units up

Answer

The graph of

y=-2 p\left[\frac{1}{2}(x-2)\right]+2

is the graph of

p(x)

scaled horizontally by 2, shifted 2 units to the right, reflected across the x-axis, scaled vertically by 2, and shifted 2 units up.

Key Concept

Transformations of functions

Explanation

Combining horizontal and vertical transformations results in the new graph.

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