Question
Question Given the definitions of \( f(x) \) and \( g(x) \) below, find the value of \( (g \circ f)(-2) \). \[ \begin{array}{l}f(x)=2 x^{2}+4 x+2 \\ g(x)=2 x-7\end{array} \]
Ask by Reese Whittaker. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\((g \circ f)(-2) = -3\)
Solution
1. Compute \( f(-2) \):
\[
f(-2) = 2(-2)^2 + 4(-2) + 2
\]
First, calculate \((-2)^2\):
\[
(-2)^2 = 4
\]
Then substitute into the expression:
\[
f(-2) = 2(4) - 8 + 2 = 8 - 8 + 2 = 2
\]
2. Compute \( g(f(-2)) \) using \( f(-2) = 2 \):
\[
g(2) = 2(2) - 7 = 4 - 7 = -3
\]
Thus, the value of \((g \circ f)(-2)\) is \(-3\).
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Simplify this solution The Deep Dive
To find the value of \( (g \circ f)(-2) \), we first need to calculate \( f(-2) \). Using the definition of \( f(x) \): \[ f(x) = 2x^2 + 4x + 2 \] Substituting \( -2 \) into \( f \): \[ f(-2) = 2(-2)^2 + 4(-2) + 2 = 2(4) - 8 + 2 = 8 - 8 + 2 = 2 \] Now that we have \( f(-2) = 2 \), we can calculate \( g(f(-2)) \) or \( g(2) \) using the definition of \( g(x) \): \[ g(x) = 2x - 7 \] Substituting \( 2 \) into \( g \): \[ g(2) = 2(2) - 7 = 4 - 7 = -3 \] Thus, the value of \( (g \circ f)(-2) \) is: \[ \boxed{-3} \]
