Question

Determine the formula for the inverse of the function\(f(x) = {x^4} + 1,x \ge 0\). Sketch the graph of \(f,{f^{ - 1}}\) and \(y = x\), check whether graphs \(f\) and \({f^{ - 1}}\) reflects about the line \(y = x\).

Short answer

The formula for the inverse of the function \(f(x) = {x^4} + 1\) is \({f^{ - 1}}(x) = \sqrt(4){{x - 1}},x \ge 0\).

Step-by-step solution

Given data

The given function is \(y = {x^4} + 1\).

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Solve the equation

Solve this equation for \(x\) as shown below.

\(\begin{array}{c}y = {x^4} + 1\\y - 1 = {x^4}\\x = \pm \sqrt(4){{y - 1}}\end{array}\)

Then, interchange \(x\) and \(y\), obtain the inverse function, \(y = \sqrt(4){{x - 1}}\) and \(y = x\)on the same axis.

From Figure, it is observed that the graph of \({f^{ - 1}}(x) = \sqrt(4){{x - 1}}\) is a reflection of the graph \({f^{ - 1}}(x) = \sqrt(4){{x - 1}}\) about the line \(y = x\).