Question
Express your answer as a polynomial in standard form. \[ \begin{array}{l}f(x)=-3 x+12 \\ g(x)=3 x^{2}-6 x-6 \\ \text { Find: }(f \circ g)(x)\end{array} \] Answer Attempt out of 2
Ask by George O'Quinn. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (f \circ g)(x) = -9x^2 + 18x + 30 \)
Solution
To find the composition of functions \( f \circ g \), we need to substitute the expression for \( g(x) \) into \( f(x) \) and simplify.
Given:
\[ f(x) = -3x + 12 \]
\[ g(x) = 3x^2 - 6x - 6 \]
Substitute \( g(x) \) into \( f(x) \):
\[ (f \circ g)(x) = f(g(x)) = -3(g(x)) + 12 \]
Substitute the expression for \( g(x) \) into the equation:
\[ (f \circ g)(x) = -3(3x^2 - 6x - 6) + 12 \]
Simplify the expression:
\[ (f \circ g)(x) = -9x^2 + 18x + 18 + 12 \]
\[ (f \circ g)(x) = -9x^2 + 18x + 30 \]
Therefore, the composition of functions \( f \circ g \) is \( -9x^2 + 18x + 30 \).
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Simplify this solution Bonus Knowledge
To find \((f \circ g)(x)\), we need to substitute \(g(x)\) into \(f(x)\). 1. First, we compute \(g(x)\): \[ g(x) = 3x^2 - 6x - 6. \] 2. Next, substitute \(g(x)\) into \(f(x)\): \[ f(g(x)) = f(3x^2 - 6x - 6). \] Since \(f(x) = -3x + 12\), we replace \(x\) with \(3x^2 - 6x - 6\): \[ f(g(x)) = -3(3x^2 - 6x - 6) + 12. \] 3. Now simplify this expression: \[ = -9x^2 + 18x + 18 + 12, \] \[ = -9x^2 + 18x + 30. \] 4. Finally, write the polynomial in standard form: \[ (f \circ g)(x) = -9x^2 + 18x + 30. \] So, the answer is: \[ (f \circ g)(x) = -9x^2 + 18x + 30. \]
