Question

Danika concludes that the following functions are inverses of each other because f(g(x)) = x. Do you agree with Danika? Explain your reasoning.
f(x) = |x|
g(x) = –x

Answer 1

that Danika needs to find g compose f

that g(f(x)) must also equal x

that g(f(x)) = –x

Answer 2

We cannot agree with Danika. Why? Well, The reasoning is given as follows:

Two functions are inverses of each other if and only if it is true that the composition function is given by:

$f(g(x))=x$

Everything is ok up to this point, right?. But let's prove that this is not fulfilled for these functions, then:

$f(x)=\left | x \right | \\ \\ g(x)=-x \\ \\ f(g(x))=\left| -x \right |=\left | -1 \right | \left | x \right |=\left | x \right | \\ \\ \therefore f(g(x))=\left | x \right | \neq x$

As you can see we did not obtain the function that matches the definition of inverse functions. For that reason we can't agree with Danika.