Look at the graph of [math]f(x)=x^3[/math] and [math]g(x)=x^1/3[/math] below. Since [math]f(g(x))=\left(g\left(x\right)\right)^3=\left(\sqrt[3]{x}\right)^3=x[/math] and [math]g(f(x))=\sqrt[3]{f\left(x\right)}=\sqrt[3]{x^3}=x[/math], they are inverse functions. What is the relationship between the graphs of the functions?

The graphs of [math]f\left(x\right)=\left(x-3\right)^3[/math] and [math]g\left(x\right)=\sqrt[3]{x}+3[/math] below also have a relationship. What does this tell you about the graphs of inverse functions?

Based on the previous graphs and the graphs of [math]f\left(x\right)=\frac{2}{x+1}[/math] and [math]g\left(x\right)=\frac{2-x}{x}[/math] below, can you make a general conjecture about the graph of a function and its inverse?

Test your conjecture by inputting functions in the graph below and comparing the resulting curves.[br][br]If you need help inputting function, please check [url=https://tube.geogebra.org/material/simple/id/1585171?doneurl=https%3A%2F%2Ftube.geogebra.org%2Fcsamaral]this page[/url].

The graph below shows the function [math]h\left(x\right)=\frac{x^3+1}{3}[/math], can you find its inverse? Input the inverse you found in the box to the left of the graph and check if the graph is the reflection of h(x) in the y=x line.

Note, however, that not always the reflection of [math]f[/math] in the line [math]y=x[/math] will be the inverse function of [math]f[/math]. To have an inverse, a function must satisfy two requirements: be injective(one-to-one) and surjective(onto), i.e. it must be a [i]bijection[/i].