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1. Determine the inverse of each function. a)

f(x)=3 x+4

g(x)=x^{3}-1

...

Apr 28, 2024

1. Determine the inverse of each function. a)

f(x)=3 x+4

g(x)=x^{3}-1

h(x)=-x

m(x)=-2(x+5)

2. Given the function

h(x)=2 x+7

, find a)

h(3)

h^{-1}(9)

Solution by Steps

step 1

To find the inverse of the function

f(x)=3x+4

, we replace

f(x)

with

y

to get

y=3x+4

step 2

We then swap

x

and

y

to get

x=3y+4

step 3

Solve for

y

to get the inverse function:

y=\frac{x-4}{3}

step 4

For the function

g(x)=x^3-1

, we start by replacing

g(x)

with

y

to get

y=x^3-1

step 5

Swap

x

and

y

to get

x=y^3-1

step 6

Solve for

y

to get the inverse function:

y=\sqrt[3]{x+1}

step 7

For the function

h(x)=-x

, we replace

h(x)

with

y

to get

y=-x

step 8

Swap

x

and

y

to get

x=-y

step 9

Solve for

y

to get the inverse function:

y=-x

step 10

For the function

m(x)=-2(x+5)

, we replace

m(x)

with

y

to get

y=-2(x+5)

step 11

Swap

x

and

y

to get

x=-2(y+5)

step 12

Solve for

y

to get the inverse function:

y=\frac{-x}{2}-5

Answer

f^{-1}(x)=\frac{x-4}{3}

, c)

g^{-1}(x)=\sqrt[3]{x+1}

, b)

h^{-1}(x)=-x

, d)

m^{-1}(x)=\frac{-x}{2}-5

Key Concept

Finding the inverse of a function

Explanation

To find the inverse of a function, we replace the function notation with

y

, swap

x

and

y

, and then solve for

y

Solution by Steps

step 1

To find

h(3)

for the function

h(x)=2x+7

, we substitute

x

with 3

step 2

Calculate the value:

h(3)=2(3)+7

step 3

Simplify to get the result:

h(3)=6+7=13

step 4

To find

h^{-1}(9)

, we start by replacing

h(x)

with

y

to get

y=2x+7

step 5

Swap

x

and

y

to get

x=2y+7

step 6

Solve for

y

to get the inverse function:

y=\frac{x-7}{2}

step 7

Substitute

x

with 9 in the inverse function:

y=\frac{9-7}{2}

step 8

Simplify to get the result:

y=\frac{2}{2}=1

Answer

h(3)=13

, b)

h^{-1}(9)=1

Key Concept

Evaluating functions and their inverses

Explanation

To evaluate a function at a given point, substitute the point into the function. To find the inverse value, solve the inverse function for the given output.