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If a number of transformations are applied to the function f(x)=x+4f(x)=\sqrt{x+4} to ...
Mar 20, 2024
If a number of transformations are applied to the function f(x)=x+4f(x)=\sqrt{x+4} to give g(x)=−3f(2x−1)g(x)=-3 f(2 x-1) then dom⁡(g)=[a,∞)\operatorname{dom}(g)=[a, \infty). What is the value of aa ? Write your answer to one decimal place.
Solution by Steps
step 1
To find the domain of g(x) g(x) , we need to determine the values of x x for which g(x) g(x) is defined
step 2
The function g(x) g(x) is given by g(x)=−3f(2x−1) g(x) = -3 f(2x - 1) , where f(x)=x+4 f(x) = \sqrt{x + 4}
step 3
The domain of f(x) f(x) is x≥−4 x \geq -4 as per the asksia-ll calculator result
step 4
To find the domain of g(x) g(x) , we need to find the domain of f(2x−1) f(2x - 1) . This means we need to solve 2x−1+4≥0 2x - 1 + 4 \geq 0
step 5
Solving the inequality 2x+3≥0 2x + 3 \geq 0 gives us x≥−32 x \geq -\frac{3}{2}
step 6
Therefore, the domain of g(x) g(x) is x≥−32 x \geq -\frac{3}{2}
Answer
a=−1.5 a = -1.5
Key Concept
Domain of a Composite Function
Explanation
The domain of the composite function g(x) g(x) is determined by the domain of the inner function f(2x−1) f(2x - 1) , which must be non-negative since it is under a square root.
The distance that a car, in peak hour traffic, is from a set of traffic lights over time is recorded. The distance dd (in metres) from the traffic lights is a function of time tt (in minutes). \begin{tabular}{|c|c|c|r|r|} \hlinett & 3 & 6 & 7 & 9 \\ \hlinedd & 28 & 22 & 16 & 4 \\ \hline \end{tabular} What is the average of the function d(t)d(t) from t=6t=6 to t=9t=9 ? Write your answer as an integer (that is, a whole number)
Solution by Steps
step 1
Calculate the sum of the distances from the traffic lights at t=6t=6, t=7t=7, and t=9t=9
step 2
Using the given distances, the sum is 22+16+422 + 16 + 4
step 3
Simplify the sum to get 4242
step 4
Divide the sum by the number of terms, which is 33, to find the average distance
step 5
The average distance is 423\frac{42}{3}
step 6
Simplify the fraction to get 1414
Answer
14
Key Concept
Average of a function over an interval
Explanation
The average value of a function over an interval is found by summing the function values at specific points and dividing by the number of points.
Suppose the point PP at (x,y)=(2,1)(x, y)=(2,1) is on the graph of the function y=f(x)y=f(x). A number of transformations are performed on the graph of y=f(x)y=f(x) to give the graph y=−3f(12x−2)+1y=-3 f\left(\frac{1}{2} x-2\right)+1. What is the xx-coordinate of the point P\mathrm{P} after the transformations are applied? Write your answer as an integer (that is, a whole number)
Generated Graph
Solution by Steps
step 1
Identify the original coordinates of point P P before the transformation
step 2
Apply the horizontal scaling transformation by a factor of 12 \frac{1}{2} to the x x -coordinate
step 3
Apply the horizontal translation transformation by moving 2 2 units to the right
step 4
Combine the transformations to find the new x x -coordinate of point P P after the transformations
Answer
The x x -coordinate of the point P P after the transformations is 8 8 .
Key Concept
Transformation of Functions
Explanation
The x x -coordinate of point P P is transformed by first scaling by 12 \frac{1}{2} and then translating by +2 +2 . The original x x -coordinate of 2 2 becomes 1 1 after scaling and 3 3 after translating, but since the transformation inside the function is 12x−2 \frac{1}{2}x - 2 , we need to solve for x x when 12x−2=3 \frac{1}{2}x - 2 = 3 . Solving this equation gives us x=8 x = 8 .
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