Question

The functions f and g are shown. Are the functions inverses?
Use the drop-down menus to explain.
f(x) = x- 1
9 (x) =x+1
Click the arrows to choose an answer from each menu.
The functionf (g (x)) is equal to
Choose...
The function g f(x)) is equal to Choose...
The functions f(x) and g x) Choose...
V
inverses.

Answer 1

To determine if the functions \( f(x) \) and \( g(x) \) are inverses of each other, we need to check if \( f(g(x)) \) and \( g(f(x)) \) both equal \( x \).

1. **Calculating \( f(g(x)) \)**:
- Given \( f(x) = x - 1 \) and \( g(x) = x + 1 \),
\[
f(g(x)) = f(x + 1) = (x + 1) - 1 = x.
\]

2. **Calculating \( g(f(x)) \)**:
- Again, using \( f(x) = x - 1 \) and \( g(x) = x + 1 \),
\[
g(f(x)) = g(x - 1) = (x - 1) + 1 = x.
\]

Since both functions satisfy the condition \( f(g(x)) = x \) and \( g(f(x)) = x \), we can conclude:

- The function \( f(g(x)) \) is equal to **\( x \)**.
- The function \( g(f(x)) \) is equal to **\( x \)**.
- The functions \( f(x) \) and \( g(x) \) **are inverses**.

So, the final answers for the drop-down menus would be:

1. The function \( f(g(x)) \) is equal to: **\( x \)**.
2. The function \( g(f(x)) \) is equal to: **\( x \)**.
3. The functions \( f(x) \) and \( g(x) \) **are inverses**.