Question

Sketch a graph of each of the following functions. Label at least three points on each graph. Name the inverse function of each and show its graph.

$$\begin{gathered} a)y=x^3 \\ b)y=e^x \\ c)y=\sin x-\frac{\pi }{2}\le x\le \frac{\pi }{2} \\ d)y=\cos x0\le x\le \pi \end{gathered}$$

Short answer

The inverse functions can be given as

$$\begin{gathered} a)x=y^{\frac{1}{3}} \\ b)x=\ln y \\ c)x={\sin }^{-1}y \\ d)x={\cos }^{-1}y \end{gathered}$$

Step-by-step solution

Given information

We are given 4 functions

Part a) Step 1: Plot the graph

The graph of the function$y=x^3$ can be given as

Part a) Step 2: Find inverse function

We have,

$$\begin{gathered} y=x^3 \\ takingcuberoot \\ x=y^{\frac{1}{3}} \end{gathered}$$

is the required inverse function

Part b) Step 1: Plot the graph

The graph of $y=e^x$can be given as

Part b) Step 2: Find inverse function

consider

$$\begin{gathered} y=e^x \\ \ln y=x \end{gathered}$$

is the required inverse function

Part c) Step 1: Plot the graph

The graph of function in the given domain is given as

Part c) Step 2: find the inverse function

The inverse function can be given as

$$\begin{gathered} y=\sin x \\ x={\sin }^{-1}y \end{gathered}$$

Part d) Step 1: Plot the graph of the function

The graph of the function can be given as

Part d) Step 2: Find the inverse function

The inverse function can be given as

$$\begin{gathered} y=\cos x \\ x={\cos }^{-1}y \end{gathered}$$