Question

Find the inverse of \(f .\) Then sketch the graphs of \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{3 / 5}+1 $$

Short answer

The inverse of the function \(f(x) = x^{\frac{3}{5}} + 1\) is \(f^{-1}(x) = (x - 1)^{\frac{5}{3}}\). To graph both functions, choose a few values of \(x\), compute the corresponding \(y\) values, and plot the points on the same axes. Draw a smooth curve through the points for each function. The graphs of \(f(x)\) and \(f^{-1}(x)\) will be symmetric with respect to the line \(y = x\).

Step-by-step solution

Set y equal to the function

We first need to set \(y\) equal to the function: \[y = x^{\frac{3}{5}} + 1\]

Swap x and y

Now we will swap \(x\) and \(y\) in the equation. This reverses the roles of the input and output: \[x = y^{\frac{3}{5}} + 1\]

Solve for y

We need to solve for \(y\) in the above equation to find the inverse function. To do this, first, we need to isolate the term containing \(y\). Subtract 1 from both sides of the equation: \[x - 1 = y^{\frac{3}{5}}\] Now we raise both sides to the power of \(\frac{5}{3}\), in order to remove the exponent \(\frac{3}{5}\): \[\left(x - 1\right)^{\frac{5}{3}} = y\] Now we found our inverse function, which is: \[f^{-1}(x) = \left(x - 1\right)^{\frac{5}{3}}\]

Graph the functions

We now need to graph both the original function \(f(x) = x^{\frac{3}{5}} + 1\) and the inverse function \(f^{-1}(x) = (x - 1)^{\frac{5}{3}}\) on the same set of axes. 1. Choose a few values of x for each function and calculate their corresponding \(y\) values. 2. Plot those points on the graph. 3. Draw a smooth curve through those points for each function. When graphing, it's helpful to know that for all inverse functions, they are reflected over the line \(y = x\). Therefore, when you graph both functions, you should notice that they are symmetric with respect to the line \(y = x\).