Question
For the functions \( f(x)=2 x^{2}-3 x-8 \), and \( g(x)=x-3 \), find \( (f \circ g)(x) \) a) \( 2 x^{2}-15 x+19 \) b) \( 2 x^{2}-12 x+9 \) c) \( 2 x^{2}-12+19 \) d) \( 2 x^{2}-6 x+10 \) e) \( 2 x^{2}-3 x-11 \)
Ask by Edwards Wright. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (f \circ g)(x) = 2x^2 - 15x + 19 \), so the answer is option (a).
Solution
We are given:
\[
f(x)=2x^2-3x-8,\quad g(x)=x-3.
\]
To find \((f \circ g)(x)\), we substitute \(g(x)\) into \(f(x)\):
\[
(f \circ g)(x)=f(g(x))=2(g(x))^2-3(g(x))-8.
\]
Since \(g(x)=x-3\), we have:
\[
(f \circ g)(x)=2(x-3)^2-3(x-3)-8.
\]
**Step 1: Expand \((x-3)^2\)**
\[
(x-3)^2 = x^2 - 6x + 9.
\]
**Step 2: Substitute back and expand**
\[
2(x-3)^2 = 2(x^2-6x+9)=2x^2-12x+18.
\]
\[
-3(x-3) = -3x+9.
\]
Thus,
\[
(f \circ g)(x)=2x^2-12x+18-3x+9-8.
\]
**Step 3: Combine like terms**
- \(x^2\) terms: \(2x^2\).
- \(x\) terms: \(-12x-3x=-15x\).
- Constant terms: \(18+9-8=19\).
So, we get:
\[
(f \circ g)(x)=2x^2-15x+19.
\]
The correct answer is option (a).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Beyond the Answer
To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). First, we calculate \( g(x) = x - 3 \). Next, we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(x - 3) = 2(x - 3)^2 - 3(x - 3) - 8 \] Now, expand \( (x - 3)^2 \): \[ (x - 3)^2 = x^2 - 6x + 9 \] Then, we can substitute back into \( f(x) \): \[ f(x - 3) = 2(x^2 - 6x + 9) - 3(x - 3) - 8 \] Now, distribute and simplify: \[ = 2x^2 - 12x + 18 - 3x + 9 - 8 \] \[ = 2x^2 - 15x + 19 \] Thus, \( (f \circ g)(x) = 2x^2 - 15x + 19 \). So, the answer is: a) \( 2 x^{2}-15 x+19 \)
