Question

Determine whether f(x)=5x+1x and g(x)=x5x+1
are inverse functions. Explain how you know.

(3 points)

Answer 1

To determine if the functions \( f(x) = 5x + 1 \) and \( g(x) = x + 5 \) are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

1. **Calculate \( f(g(x)) \)**:

\[
g(x) = x + 5
\]
Now substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(x + 5) = 5(x + 5) + 1 = 5x + 25 + 1 = 5x + 26
\]

Since \( f(g(x)) = 5x + 26 \neq x \), the functions \( f \) and \( g \) do not satisfy the first condition for being inverse functions.

2. **Calculate \( g(f(x)) \)**:

Now let's calculate \( g(f(x)) \):

\[
f(x) = 5x + 1
\]
Substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(5x + 1) = (5x + 1) + 5 = 5x + 1 + 5 = 5x + 6
\]

Since \( g(f(x)) = 5x + 6 \neq x \), the functions \( f \) and \( g \) fail the second condition as well.

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \) holds true, we conclude that \( f(x) \) and \( g(x) \) are **not inverse functions**.