Question

Molly is verifying if the two functions are inverses of each other? Her answer is as follows:

f(g(x))=x+5 g(f(x))=x+5

She stated that they are not inverses of each other. Is she correct and why? Explain.

Answer 1

Molly is correct. The fact that $f(g(x)) = g(f(x)) = x+5$ indicates that the two functions, $f$ and $g$, are symmetric about the line $y=x$, which means that they are not inverses of each other.

To determine whether two functions are inverses of each other, we need to show that their composition results in the identity function. That is, if $f(x)$ and $g(x)$ are two functions, then $f(g(x)) = g(f(x)) = x$ for all $x$ in the domain of $f$ and $g$.

In this case, we see that $f(g(x)) = g(f(x)) = x+5$, which is not the identity function. Therefore, the two functions are not inverses of each other.

Answer 2

Molly is correct in stating that the two functions are not inverses of each other.

To be inverses of each other, two functions must satisfy the property that when they are composed in either order, they result in the identity function, which is represented by f(x) = x.

In this case, we have:

f(g(x)) = (x + 5) + 5 = x + 10

g(f(x)) = (x + 5) + 5 = x + 10

Since both compositions of the functions result in x + 10, which is not equal to x, the two functions are not inverses of each other.

Explanation: