Calculating Inverse function

Question

Please help me with the following question: Calculate, if possible, the inverse of the following functions:

(i) $f(x) = (2x - 2)^5$

(ii) $f(x) = (2x - 3)/4$

(iii) $f(x) = x^2 + 1,$ for $ x \geq 0$.

Answer 1

For each of your functions, ensure that they are bijective, assuming the first two functions are such that $f:\mathbb R \to \mathbb R$, and the third function, restricted to $f: \mathbb R_{x\geq 0} \to \mathbb R_{x \geq 0}$.

Your functions are bijective, as so defined, but it's important to know that an inverse function exists if and only if a function is bijective, or more correctly, we can find an inverse function when the domain and codomain are limited in a way so that the inverse of a function can be defined, when that's possible.

I'll help you along with the first question, using a standard method you can use for most problems of this type.

$$f(x) = y =(2x - 2)^5$$

(i) Solve for $x$: Express the function as a function of $y$.

$$\begin{align} y = (2x - 2)^5 &\iff \sqrt[\large 5] y = 2x - 2 = 2(x - 1) \\ \\ &\iff \frac{\sqrt[\large 5] y}{2} = x - 1 \\ \\ & \iff x = \frac{\sqrt[\large 5] y}{2} + 1\end{align}$$

(ii) Now, exchange $x$ for $y$: $$f^{-1}(x) = y = \frac{\sqrt[\large 5] x}{2} + 1$$

(iii) Double check to ensure that $f(f^{-1}(x)) = x$, since the composition of a function and its inverse is equivalent to the identity function that maps x to x.