Question

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

Answer 1

graph{-x^3+1 [-10, 10, -5, 5]}

Explanation:

Ok so first let's look at the parent function of the cube root --> x^3.
From there what are the transformations? Let's list them:

1. Reflection over the x-axis
2. one unit up

If those are your transformations, then you are well prepared to graph the function. Why not go through the steps?

Step 1: imagine the parent function of the cube root
Step 2: reflect the parent function over the x-axis
Step 3: move the vertex up one unit ( the vertex is 0,1 for this equation )

Now that we have the graph, we should test the inverse. One way to make sure that you have the inverses is f(g(x))=x and g(f(x))=x. But with the horizontal line test, all the graph has to do is to cross/touch once. If it crosses twice, then it is not a function.

Just so you know, the vertical line test is to test if the function works and the horizontal line test is to see if the inverse will work.

I hope that this helps and good days to you!