3.8E: Inverses and Radical Functions (Exercises)
section 3.8 exercise
For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain.
1. \(f\left(x\right)=\left(x-4\right)^{2}\)
2. \(f\left(x\right)=\left(x+2\right)^{2}\)
3. \(f\left(x\right)=12-x^{2}\)
4. \(f\left(x\right)=9-x^{2}\)
5. \(f\left(x\right)=3x^{3} +1\)
6. \(f\left(x\right)=4-2x^{3}\)
Find the inverse of each function.
7. \(f\left(x\right)=9+\sqrt{4x-4}\)
8. \(f\left(x\right)=\sqrt{6x-8} +5\)
9. \(f\left(x\right)=9+2\sqrt[{3}]{x}\)
10. \(f\left(x\right)=3-\sqrt[{3}]{x}\)
11. \(f\left(x\right)=\dfrac{2}{x+8}\)
12. \(f\left(x\right)=\dfrac{3}{x-4}\)
13. \(f\left(x\right)=\dfrac{x+3}{x+7}\)
14. \(f\left(x\right)=\dfrac{x-2}{x+7}\)
15. \(f\left(x\right)=\dfrac{3x+4}{5-4x}\)
16. \(f\left(x\right)=\dfrac{5x+1}{2-5x}\)
Police use the formula \(v=\sqrt{20L}\) to estimate the speed of a car, \(v\), in miles per hour, based on the length, \(L\), in feet, of its skid marks when suddenly braking on a dry, asphalt road.
17. At the scene of an accident, a police officer measures a car’s skid marks to be 215 feet long. Approximately how fast was the car traveling?
18. At the scene of an accident, a police officer measures a car’s skid marks to be 135 feet long. Approximately how fast was the car traveling?
The formula \(v=\sqrt{2.7r}\) models the maximum safe speed, \(v\), in miles per hour, at which a car can travel on a curved road with radius of curvature \(r\), in feet.
19. A highway crew measures the radius of curvature at an exit ramp on a highway as 430 feet. What is the maximum safe speed?
20. A highway crew measures the radius of curvature at a tight corner on a highway as 900 feet. What is the maximum safe speed?
21. A drainage canal has a cross-section in the shape of a parabola. Suppose that the canal is 10 feet deep and 20 feet wide at the top. If the water depth in the ditch is 5 feet, how wide is the surface of the water in the ditch? [UW]
22. Brooke is located 5 miles out from the nearest point \(A\) along a straight shoreline in her sea kayak. Hunger strikes and she wants to make it to Kono’s for lunch; see picture. Br ooke can paddle 2 mph and walk 4 mph. [UW]
a. If she paddles along a straight line course to the shore, find an expression that computes the total time to reach lunch in terms of the location where Brooke beaches her kayak.
b. Determine the total time to reach Kono’s if she paddles directly to the point \(A\).
c. Determine the total time to reach Kono’s if she paddles directly to Kono’s.
d. Do you think your answer to b or c is the minimum time required for Brooke to reach lunch?
e. Determine the total time to reach Kono’s if she paddles directly to a point on the shore half way between point A and Kono’s. How does this time compare to the times in parts b or c? Do you need to modify your answer to part d?
23. Clovis is standing at the edge of a dropoff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing. With the origin of the coordinate system located where he is standing, and the \(x\)-axis extending horizontally, the path of the rocket is described by the formula \(y=-2x^{2} +120x\). [UW]
a. Give a function \(h=f(x)\) relating the height \(h\) of the rocket above the sloping ground to its \(x\)-coordinate.
b. Find the maximum height of the rocket above the sloping ground. What is its \(x\)-coordinate when it is at its maximum height?
c. Clovis measures the height \(h\) of the rocket above the sloping ground while it is going up. Give a function \(x=g\left(h\right)\) relating the \(x\)-coordinate of the rocket to \(h\).
d. Does the function from (c) still work when the rocket is going down? Explain.
24 . A trough has a semicircular cross section with a radius of 5 feet. Water starts flowing into the trough in such a way that the depth of the water is increasing at a rate of 2 inches per hour. [UW]
a. Give a function \(w=f\left(t\right)\) relating the width w of the surface of the water to the time \(t\), in hours. Make sure to specify the domain and compute the range too.
b. After how many hours will the surface of the water have width of 6 feet?
c. Give a function \(t=f^{-1} \left(w\right)\) relating the time to the width of the surface of the water. Make sure to specify the domain and compute the range too.
- Answer
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1. Domain \((4, \infty)\) Inverse \(f^{-1} (x) = \sqrt{x} + 4\)
3. Domain \((-\infty, 0)\) Inverse \(f^{-1} (x) = -\sqrt{12 - x}\)
5. Domain \((-\infty, 0)\) Inverse \(f^{-1} (x) = \sqrt[3]{\dfrac{x - 1}{3}}\)
7. \(f^{-1} (x) = \dfrac{(x - 9)^2}{4} + 1\)
9. \(f^{-1} (x) = (\dfrac{x - 9}{2})^3\)
11. \(f^{-1} (x) = \dfrac{2 - 8x}{x}\)
13. \(f^{-1} (x) = \dfrac{3 - 7x}{x - 1}\)
15. \(f^{-1} (x) = \dfrac{5x - 4}{3 + 4x}\)
17. 65.574 mph
19. 34. 073 mph
21. 14.142 feet