Question
Using the following equations, find \( g(f(x)) \) : \( f(x)=2 x^{2}+4 \) \( g(x)=8 x+7 \) \( g(f(x))=[?] x^{2}+\square \)
Ask by Rowe Wells. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( g(f(x)) = 16x^{2} + 39 \)
Solution
To find \( g(f(x)) \), we need to compose the functions \( f \) and \( g \). Here's how to do it step by step:
1. **Given Functions:**
\[
f(x) = 2x^{2} + 4
\]
\[
g(x) = 8x + 7
\]
2. **Compose \( g(f(x)) \):**
\[
g(f(x)) = g(2x^{2} + 4)
\]
3. **Substitute \( f(x) \) into \( g(x) \):**
\[
g(2x^{2} + 4) = 8(2x^{2} + 4) + 7
\]
4. **Expand and Simplify:**
\[
8(2x^{2} + 4) + 7 = 16x^{2} + 32 + 7 = 16x^{2} + 39
\]
So, the composed function \( g(f(x)) \) is:
\[
g(f(x)) = 16x^{2} + 39
\]
**Final Answer:**
\[
g(f(x)) = 16\,x^{2} +\,39
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To find \( g(f(x)) \), we start by substituting \( f(x) \) into \( g(x) \). First, we have: \[ f(x) = 2x^2 + 4. \] Now, replace \( x \) in \( g(x) \) with \( f(x) \): \[ g(f(x)) = g(2x^2 + 4) = 8(2x^2 + 4) + 7. \] Now, distribute \( 8 \): \[ g(f(x)) = 16x^2 + 32 + 7 = 16x^2 + 39. \] So, we have: \[ g(f(x)) = 16 x^{2} + 39. \]
