Question

How do you graph `f(x)=1/4x^3` and then use the horizontal test to determine whether the inverse of f is a function?

Answer 1

Graphing calculator or transformation
See if any horizontal line passes through more than one point on the graph.

Explanation:

There are many ways to graph it. Obviously, you can use a graphing calculator (Desmos, etc.):

graph{(1/4)(x^3) [-10, 10, -5, 5]}

Or, you can use a `f(x)=x^3` graph and vertically shrink it by a factor of 4.

The horizontal line test works like this: If you can draw a horizontal line anywhere on the graph and it only passes through one point of the function, then the function has an inverse funtion. It seems that this graph passes the horizontal line test since any horizontal line passes through only one point. An example of a graph that does not pass the horizontal line test is `f(x)=x^2`.

If you want to know why the horizontal line test works or if you need help with finding the inverse function of this graph, just comment! Hope this helps!