Question

How do you graph the function `f(x)=x^3-2` and its inverse?

Answer 1

See explanation

Explanation:

To graph `f(x)=x^3-2` it is helpful to remember what the graph of `f(x)=x^3` looks like (See below):

`f(x)=x^3`

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

To graph `f(x)=x^3-2`, all we really doing is shifting the `x`-values down `2` units.

`f(x)=x^3-2`

graph{x^3-2 [-10, 10, -5, 5]}

As for graphing we first must find the inverse, `f^(-1)(x)`. We do this by switching the roles of `x` and `y` and solving for #y.

We can rewrite `f(x)=x^3-2` as `y=x^3-2` which then becomes `x=y^3-2` when we switch the roles of `x` and `y`

Now, we solve for `y`

`x=y^3-2`

`x+2=y^3+cancel(-2+2`

`root(3)(x+2)=root(3)(y^3)`

`root(3)(x+2)=y`

Our inverse function can then be rewritten as:

`f^(-1)(x)=root(3)(x+2)`

To graph `f^(-1)(x)=root(3)(x+2)`, take the function of `f(x)=root(3)x` and shift the `x`-values `2` units to the left:

`f(x)=root(3)(x+2)`

graph{root(3)(x+2) [-10, 10, -5, 5]}

The graph of the function is below along with the original function:

graph{(y-x^3+2)(y-root(3)(x+2))=0 [-10, 10, -5, 5]}