Question

Determine whether the functions are inverses1. f(×) = 2(×-4)2. f(×)=(×-1)square + 2

Answer 1

Given the two functions below

$\begin{gathered} f(x)=2(x-4) \\ f(x)=(x-1)^2+2 \end{gathered}$

In other to determine whether the functions are inverse, we would find the inverse of both functions as shown below

$\begin{gathered} f(x)=2(x-4) \\ \frac{f(x)}{2}=x-4 \\ \frac{f(x)}{2}+4=x \\ \text{replace f(x) with x},\text{ and x with f'(x)} \\ \\ \frac{x}{2}+4=f^{\prime}(x) \\ f^{\prime}(x)=\frac{x}{2}+4 \end{gathered}$$\begin{gathered} f(x)=(x-1)^2+2 \\ f(x)-2=(x-1)^2 \\ \sqrt[]{f(x)-2}=x-1 \\ \sqrt[]{f(x)-2}+1=x \\ \text{replace f(x) with x},\text{ and x with f'(x)} \\ \sqrt[]{x-2}+1=f^{\prime}(x) \\ f^{\prime}(x)=\sqrt[]{x-2}+1 \end{gathered}$

It can be observed from the inverse function that none of the inverse functions is equal to the original function of the given question

Hence, the functions are not inverses