Question
\[ f(x) = 3x + 4 \quad g(x) = 2x - 3 \] \[ \text{Find: } f \circ g(x) \]
Ask by Santiago Graham. in Kenya
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f \circ g(x) = 6x - 5 \)
Solution
1. We are given the functions:
\[
f(x)=3x+4 \quad \text{and} \quad g(x)=2x-3.
\]
2. The composition \( f \circ g \) is defined as:
\[
(f \circ g)(x)=f(g(x)).
\]
3. Substitute \( g(x)=2x-3 \) into \( f(x) \):
\[
f(g(x))=f(2x-3)=3(2x-3)+4.
\]
4. Distribute \( 3 \) inside the parentheses:
\[
3(2x-3)=6x-9.
\]
5. Add \( 4 \) to obtain the final expression:
\[
6x-9+4=6x-5.
\]
6. Therefore, the composition is:
\[
(f \circ g)(x)=6x-5.
\]
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Simplify this solution Beyond the Answer
To find \( f \circ g(x) \), we need to substitute \( g(x) \) into \( f(x) \). 1. First, we find \( g(x) \): \[ g(x) = 2x - 3 \] 2. Next, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(2x - 3) \] 3. Now, apply the function \( f \): \[ f(x) = 3x + 4 \quad \Rightarrow \quad f(2x - 3) = 3(2x - 3) + 4 \] 4. Simplifying: \[ = 6x - 9 + 4 = 6x - 5 \] Therefore, \( f \circ g(x) = 6x - 5 \).
