Chapter 4.7, Problem 20PPS

a.

To determine

To find: The inverse of the function $C(x)$ .

Answer to Problem 20PPS

  $C^{-1}(x)=\frac{x-50}{12}$

Explanation of Solution

Given information: The function for total monthly cost is $C(x)=50+12x$

Calculation:

Let the $C(x)$ as y then rewrite the function as $y=50+12x$

Now, swap the variables,

  $x=50+12y$

Solve it for y to get inverse function.

Subtract both sides 50.

  $x-50=12y$

Divide both sides by 12.

  $\frac{x-50}{12}=\frac{12y}{12}$

Simplify it,

  $y=\frac{x-50}{12}$

So, the inverse function is (replace y as $C^{-1}(x)$ )

  $C^{-1}(x)=\frac{x-50}{12}$

b.

To determine

To find: The quantities represented by x and $C^{-1}(x)$ in the context of inverse function.

Answer to Problem 20PPS

In the inverse function, x represents total monthly cost and $C^{-1}(x)$ represents number of hours spent riding.

Explanation of Solution

Given information: A function for total monthly cost $C(x)=50+12x$ where x represents the number of hours spent on riding.

While finding the inverse of a function all x are replaced as $C(x)$ and all $C(x)$ as x. Thus it means the variables x and $C(x)$ are interchanged. Then equation is solved for $C(x)$ and $C(x)$ is replaced as $C^{-1}(x)$ .

c.

To determine

To find: The total number of hours spent on riding for the total cost is $338.

Answer to Problem 20PPS

Number of hours spent is 24 hours.

Explanation of Solution

Given information: In this context $C(x)$ is $338

Calculation:

Plug in $C(x)$ as $338 then solve it for ‘x’

  $338=50+12x$

Subtract both sides 50,

  $288=12x$

Divide both sides by 12.

  $x=24$

Conclusion: Number of hours spent is 24 hours.