Question
12:15 Composition of Functions Using the following equations, find \( g(f(x)) \) : \[ \begin{array}{c} f(x)=2 x \\ g(x)=9 x+3 \\ g(f(x))=[?] x+ \end{array} \]
Ask by Edwards Goodwin. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( g(f(x)) = 18x + 3 \)
Solution
To find the composition \( g(f(x)) \), we'll substitute the function \( f(x) \) into the function \( g(x) \).
**Given:**
\[
\begin{align*}
f(x) &= 2x \\
g(x) &= 9x + 3
\end{align*}
\]
**Step-by-Step Solution:**
1. **Find \( g(f(x)) \):**
\[
g(f(x)) = g(2x)
\]
2. **Substitute \( f(x) = 2x \) into \( g(x) \):**
\[
g(2x) = 9(2x) + 3
\]
3. **Simplify the expression:**
\[
9(2x) = 18x \\
18x + 3 = 18x + 3
\]
4. **Final Composition:**
\[
g(f(x)) = 18x + 3
\]
**Answer:**
\[
g(f(x)) = 18x + 3
\]
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Simplify this solution Bonus Knowledge
To find \( g(f(x)) \), we first need to substitute \( f(x) \) into \( g(x) \). Given \( f(x) = 2x \), we can plug this into \( g(x) = 9x + 3 \): 1. Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(2x) \] 2. Now, replace \( x \) in \( g(x) \) with \( 2x \): \[ g(2x) = 9(2x) + 3 = 18x + 3 \] Hence, \( g(f(x)) = 18x + 3 \). So, the answer is: \[ g(f(x)) = 18x + 3 \]
