Question

Verify the equation is correct and fill in the blanks for B

Answer 1

Given that the function

$f(x)\text{ =13x}$

is a one-to-one function,

A) Equation for f⁻¹(x), the inverse function.

From the function f(x), interchange x and f(x).

thus,

$\begin{gathered} f(x)\text{ = 13x} \\ in\text{terchanging x and f(x), we have} \\ x\text{ = 13f(x)} \\ \text{divide both sides by 13} \\ f(x)\text{ = }\frac{x}{13} \end{gathered}$

thus, equation for f⁻¹(x), the inverse function is

$f(x)\text{ = }\frac{x}{13}$

B) To show that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Starting with f(f⁻¹(x)):

Evaluating f(f⁻¹(x)) involves substituting the f⁻¹(x) function into the f(x) function.

In this case, let f⁻¹(x) be z(x).

Thus, we have f(f⁻¹(x)) to be f(z).

$\begin{gathered} z(x)=f^{-1}(x)\text{ = }\frac{x}{13} \\ f(x)\text{ = }13x \\ \text{thus, substituting the z(x) function into the f(x) function, we have} \\ f(z(x))\text{ = }13(\frac{x}{13}) \\ \Rightarrow x \\ \end{gathered}$

For f⁻¹(f(x)):

Similarly, evaluating f⁻¹(f(x)) involves substituting the f(x) function into the f⁻¹(x) function.

Thus, we have

$\begin{gathered} f(x)\text{ = 13x} \\ f^{-1}(x)\text{ = }\frac{x}{13} \\ \text{thus, substituting f(x) into }f^{-1}(x)\text{ gives} \\ f^{-1}(f(x))\text{ = }\frac{1}{13}(13x) \\ \Rightarrow x \\ \end{gathered}$

Hence,

$\begin{gathered} f(f^{-1}(x))\text{ = }f(\frac{x}{13}) \\ \text{ = x} \end{gathered}$$\begin{gathered} f^{-1}(f(x))\text{ = }f^{-1}(13x) \\ =\text{ x} \end{gathered}$