Question

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Explanation:

Question 12: Find \( f \circ g \)

Step 1: Recall the composite function formula

The formula for \( (f \circ g)(x) \) is \( f(g(x)) \). So we need to substitute \( g(x) \) into \( f(x) \).
Given \( f(x) = 3x + 5 \) and \( g(x) = x - 3 \), we substitute \( g(x) \) (which is \( x - 3 \)) in place of \( x \) in \( f(x) \).

Step 2: Substitute \( g(x) \) into \( f(x) \)

\[

$$\begin{align*} (f \circ g)(x) &= f(g(x)) \\ &= f(x - 3) \\ &= 3(x - 3) + 5 \end{align*}$$

\]

Step 3: Simplify the expression

First, distribute the 3: \( 3(x - 3) = 3x - 9 \). Then add 5:
\[

$$\begin{align*} 3x - 9 + 5 &= 3x - 4 \end{align*}$$

\]

Step 1: Recall the composite function formula

The formula for \( (g \circ f)(x) \) is \( g(f(x)) \). So we need to substitute \( f(x) \) into \( g(x) \).
Given \( f(x) = 3x + 5 \) and \( g(x) = x - 3 \), we substitute \( f(x) \) (which is \( 3x + 5 \)) in place of \( x \) in \( g(x) \).

Step 2: Substitute \( f(x) \) into \( g(x) \)

\[

$$\begin{align*} (g \circ f)(x) &= g(f(x)) \\ &= g(3x + 5) \\ &= (3x + 5) - 3 \end{align*}$$

\]

Step 3: Simplify the expression

Subtract 3 from 5: \( 5 - 3 = 2 \). So the expression becomes:
\[
3x + 2
\]

Answer:

\( (f \circ g)(x) = 3x - 4 \)

Question 13: Find \( g \circ f \)