Question

How do you find the inverse of `f(x)=x^2-2` from `x<=0` and graph both f and `f^-1`?

Answer 1

See below.

Explanation:

To find the inverse of a function, you can switch the `x` and `y` variables and solve for `y`.

`y=x^2 - 2`

`x = y^2 - 2` `->` switch variables

`x+2 = y^2`

`y = +- sqrt (x+2)`

However, since `f(x)` is restricted to `x<=0`, `f^(-1)(x)` will be restricted to `y<=0`. We only take the negative result above.

`f^(-1)(x) = -sqrt(x+2)`

To graph `f(x)`, simply graph the parabola but keep only the negative `x`-values.

To graph `f^(-1)(x)`, graph `sqrtx`, then translate it `2` units left and reflect it over the `x`-axis.

If two functions are inverses, they should be reflections of each other over the line `y=x`. The graph below confirms that they are indeed inverses.