Question:
If $f(x) = 3x + 5$ and $g(x) = x^2 - 1$, then $(f \circ g)(x^2 - 1)$ is equal to:
If $f(x) = 3x + 5$ and $g(x) = x^2 - 1$, then $(f \circ g)(x^2 - 1)$ is equal to:
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Be careful with nested substitutions. It is often safer to simplify the inner composition $f(g(x))$ first before plugging in the final variable expression.
Updated On: May 2, 2026
- $3x^4 - 3x + 5$
- $3x^4 - 6x^2 + 5$
- $6x^4 + 3x^2 + 5$
- $6x^4 - 6x + 5$
- $3x^2 + 6x + 4$
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The Correct Option is B
Solution and Explanation
Concept:
A composite function $(f \circ g)(z)$ is evaluated as $f(g(z))$. In this problem, we first determine the expression for $(f \circ g)(x)$ and then substitute $x^2 - 1$ into that result.
Step 1: Find the composite function $(f \circ g)(x)$.
\[ (f \circ g)(x) = f(g(x)) = f(x^2 - 1) \] Substitute $x^2 - 1$ into $f(x) = 3x + 5$: \[ 3(x^2 - 1) + 5 = 3x^2 - 3 + 5 = 3x^2 + 2 \]
Step 2: Substitute $(x^2 - 1)$ into the composite function.
We need to find $(f \circ g)(x^2 - 1)$. Let $u = x^2 - 1$. We found $(f \circ g)(u) = 3u^2 + 2$: \[ 3(x^2 - 1)^2 + 2 \]
Step 3: Expand the algebraic expression.
\[ 3(x^4 - 2x^2 + 1) + 2 = 3x^4 - 6x^2 + 3 + 2 \] \[ 3x^4 - 6x^2 + 5 \]
Step 1: Find the composite function $(f \circ g)(x)$.
\[ (f \circ g)(x) = f(g(x)) = f(x^2 - 1) \] Substitute $x^2 - 1$ into $f(x) = 3x + 5$: \[ 3(x^2 - 1) + 5 = 3x^2 - 3 + 5 = 3x^2 + 2 \]
Step 2: Substitute $(x^2 - 1)$ into the composite function.
We need to find $(f \circ g)(x^2 - 1)$. Let $u = x^2 - 1$. We found $(f \circ g)(u) = 3u^2 + 2$: \[ 3(x^2 - 1)^2 + 2 \]
Step 3: Expand the algebraic expression.
\[ 3(x^4 - 2x^2 + 1) + 2 = 3x^4 - 6x^2 + 3 + 2 \] \[ 3x^4 - 6x^2 + 5 \]
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