Find f[g(x)] and g[f(x)] for the given functions. 3 f(x) = -x³ -x+2, g(x) = 3x + 5 f[g(x)] = (Simplify your answer. Do not factor.)
Find f[g(x)] and g[f(x)] for the given functions. 3 f(x) = -x³ -x+2, g(x) = 3x + 5 f[g(x)] = (Simplify your answer. Do not factor.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem:**
Find \( f[g(x)] \) and \( g[f(x)] \) for the given functions.
Given:
\[ f(x) = -x^3 + 2, \]
\[ g(x) = 3x + 5. \]
**Task:**
Calculate \( f[g(x)] = \) (Simplify your answer. Do not factor.)
**Solution:**
To find \( f[g(x)] \), substitute \( g(x) = 3x + 5 \) into the function \( f(x) = -x^3 + 2 \).
So, \( f[g(x)] = f(3x + 5) \).
Replace \( x \) in \( f(x) \) with \( 3x + 5 \):
\[ f(3x + 5) = - (3x + 5)^3 + 2. \]
Now, expand \( (3x + 5)^3 \) and substitute back into the equation:
\[ (3x + 5)^3 = (3x+5)(3x+5)(3x+5). \]
Expand this expression step by step:
1. First two factors: \( (3x + 5)(3x + 5) = 9x^2 + 30x + 25. \)
2. Multiply the result by the third factor:
\[ (9x^2 + 30x + 25)(3x + 5) = 27x^3 + 45x^2 + 75x + 25x^2 + 150x + 125. \]
Combine like terms:
\[ = 27x^3 + 70x^2 + 225x + 125. \]
Substitute back to find \( f[g(x)] \):
\[ f(3x + 5) = - (27x^3 + 70x^2 + 225x + 125) + 2 \]
\[ = -27x^3 - 70x^2 - 225x - 125 + 2 \]
\[ = -27x^3 - 70x^2 - 225x - 123. \]
Thus, the simplified expression is:
\[ f[g(x)] = -27x^3 - 70x^2 - 225x - 123.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3f191a8-a1b2-42ed-ac9e-32d60645d728%2F6bd6282c-408f-4adf-b4c4-85c83040bf25%2Ff6sth8q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find \( f[g(x)] \) and \( g[f(x)] \) for the given functions.
Given:
\[ f(x) = -x^3 + 2, \]
\[ g(x) = 3x + 5. \]
**Task:**
Calculate \( f[g(x)] = \) (Simplify your answer. Do not factor.)
**Solution:**
To find \( f[g(x)] \), substitute \( g(x) = 3x + 5 \) into the function \( f(x) = -x^3 + 2 \).
So, \( f[g(x)] = f(3x + 5) \).
Replace \( x \) in \( f(x) \) with \( 3x + 5 \):
\[ f(3x + 5) = - (3x + 5)^3 + 2. \]
Now, expand \( (3x + 5)^3 \) and substitute back into the equation:
\[ (3x + 5)^3 = (3x+5)(3x+5)(3x+5). \]
Expand this expression step by step:
1. First two factors: \( (3x + 5)(3x + 5) = 9x^2 + 30x + 25. \)
2. Multiply the result by the third factor:
\[ (9x^2 + 30x + 25)(3x + 5) = 27x^3 + 45x^2 + 75x + 25x^2 + 150x + 125. \]
Combine like terms:
\[ = 27x^3 + 70x^2 + 225x + 125. \]
Substitute back to find \( f[g(x)] \):
\[ f(3x + 5) = - (27x^3 + 70x^2 + 225x + 125) + 2 \]
\[ = -27x^3 - 70x^2 - 225x - 125 + 2 \]
\[ = -27x^3 - 70x^2 - 225x - 123. \]
Thus, the simplified expression is:
\[ f[g(x)] = -27x^3 - 70x^2 - 225x - 123.
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