An inverse function, of a function [math]f[/math], is a function such that [math]f^{-1}\left(f\left(x\right)\right)=x[/math] for all x in the domain of [math]f[/math] and [math]f\left(f^{-1}\left(y\right)\right)=y[/math] for all y in the target of [math]f[/math]. [br][br]Informally we can think of [math]f^{-1}\left(x\right)[/math] undoing the work that [math]f(x)[/math] has done. Therefore, when evaluating [math]f^{-1}\circ f\left(a\right)[/math] or [math]f\circ f^{-1}\left(a\right)[/math], the original input, a, will be obtained.[br][br]The graph below shows the functions [math]f\left(x\right)=e^x[/math], [math]g(x)=ln(x)[/math] and the points A and B. By adjusting the values of [i]a[/i], you can move the points A and B. [br]
What is the relationship between the points A and B?
Calculate [math]f(a)[/math] for [math]a=2.0;a=0.5;a=-1.0[/math]. [br]
What happens if you plug [math]f(a)[/math] as an input into the function [math]g[/math] for the values of [math]a[/math] in Question 2?
If for all values [math]a[/math] in the domain of [math]f[/math] evaluating [math]f(a)[/math] gives some value [math]b[/math], and evaluating [math]g(b)[/math] returns [math]a[/math], what is the relationship between the functions?
Using the graph of [math]h\left(x\right)=x^3[/math] below, and your knowledge of inverse functions, can you estimate the values of [math]\sqrt[3]{2.2},\sqrt[3]{-6.86},\sqrt[3]{1.52}[/math]?