Questions: For f(x)=5x-9 and g(x)=(1/5)(x+9), find (f ∘ g)(x) and (g ∘ f)(x). Then determine whether (f ∘ g)(x)=(g ∘ f)(x). What is (f ∘ g)(x)?

Solution Steps

The composition of two functions \( f \) and \( g \) is denoted as \( (f \circ g)(x) \), which means \( f(g(x)) \). Similarly, \( (g \circ f)(x) \) means \( g(f(x)) \).

Given:

• \( f(x) = 5x - 9 \)
• \( g(x) = \frac{1}{5}(x + 9) \)

To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \):

\[ (f \circ g)(x) = f(g(x)) = f\left(\frac{1}{5}(x + 9)\right) \]

Substitute \( \frac{1}{5}(x + 9) \) into \( f(x) = 5x - 9 \):

\[ f\left(\frac{1}{5}(x + 9)\right) = 5\left(\frac{1}{5}(x + 9)\right) - 9 \]

Simplify the expression:

\[ = (x + 9) - 9 \]

\[ = x \]

To find \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \):

\[ (g \circ f)(x) = g(f(x)) = g(5x - 9) \]

Substitute \( 5x - 9 \) into \( g(x) = \frac{1}{5}(x + 9) \):

\[ g(5x - 9) = \frac{1}{5}((5x - 9) + 9) \]

Simplify the expression:

\[ = \frac{1}{5}(5x) \]

\[ = x \]

From the calculations above, we have:

\[ (f \circ g)(x) = x \]

\[ (g \circ f)(x) = x \]

Since both compositions result in \( x \), we conclude that \( (f \circ g)(x) = (g \circ f)(x) \).

Final Answer