Question

1. does the graph below have an inverse? why? 9. use the graph of f(x) to graph its inverse function on the same graph. 8. yes or no (circle one) why?

Explanation:

Step1: Recall inverse - function condition

A function has an inverse if and only if it is one - to - one. A one - to - one function passes the horizontal line test, which means that no horizontal line intersects the graph of the function more than once.

Step2: Apply horizontal line test

For the given graph, if we draw horizontal lines across the graph, we can see that each horizontal line intersects the graph at most once. So the function is one - to - one.

Step3: Conclusion for existence of inverse

Since the function is one - to - one, it has an inverse.

Step4: Graphing the inverse

To graph the inverse of a function \(y = f(x)\) on the same graph, we use the fact that the graph of \(y = f(x)\) and its inverse \(y = f^{-1}(x)\) are symmetric about the line \(y=x\). We can take points \((x,y)\) on the graph of \(f(x)\) and swap the \(x\) and \(y\) coordinates to get points \((y,x)\) for the graph of \(f^{-1}(x)\) and plot them.

Answer:

1. Yes. Because the function passes the horizontal line test (is one - to - one).
2. To graph the inverse, take points on \(y = f(x)\), swap \(x\) and \(y\) coordinates and plot the new points. The graph of the inverse is symmetric to the graph of \(y = f(x)\) about the line \(y = x\).