Given the functions: f(x)=x²³-8x g(x) = √2x h(x) = 6x+7 Evaluate the function h (f(x)) for x = 3. Write your answer in exact simplified form. Select "Undefined" if applicable. h (f (3)) is √ Undefined X
Given the functions: f(x)=x²³-8x g(x) = √2x h(x) = 6x+7 Evaluate the function h (f(x)) for x = 3. Write your answer in exact simplified form. Select "Undefined" if applicable. h (f (3)) is √ Undefined X
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![### Evaluating Composite Functions
Given the functions:
\[ f(x) = x^3 - 8x \]
\[ g(x) = \sqrt{2x} \]
\[ h(x) = 6x + 7 \]
Evaluate the function \( h(f(x)) \) for \( x = 3 \). Write your answer in exact simplified form. Select "Undefined" if applicable.
**Step-by-Step Solution:**
1. **Calculate \( f(3) \):**
Substitute \( x = 3 \) into the function \( f(x) \).
\[ f(3) = 3^3 - 8 \cdot 3 \]
\[ f(3) = 27 - 24 \]
\[ f(3) = 3 \]
2. **Evaluate \( h(f(x)) \) at \( x = 3 \):**
Since \( f(3) = 3 \), now substitute this value into the function \( h(x) \).
\[ h(f(3)) = h(3) \]
\[ h(3) = 6 \cdot 3 + 7 \]
\[ h(3) = 18 + 7 \]
\[ h(3) = 25 \]
Therefore, \( h(f(3)) \) is 25.
### Final Answer:
\[ h(f(3)) \text{ is } 25 \]
[ ] Undefined](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff47ffb76-5051-461b-8a6a-8f4f8799bf23%2F5171b1a7-0b7d-484a-87c9-e311f6bf2f52%2F501kcko_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating Composite Functions
Given the functions:
\[ f(x) = x^3 - 8x \]
\[ g(x) = \sqrt{2x} \]
\[ h(x) = 6x + 7 \]
Evaluate the function \( h(f(x)) \) for \( x = 3 \). Write your answer in exact simplified form. Select "Undefined" if applicable.
**Step-by-Step Solution:**
1. **Calculate \( f(3) \):**
Substitute \( x = 3 \) into the function \( f(x) \).
\[ f(3) = 3^3 - 8 \cdot 3 \]
\[ f(3) = 27 - 24 \]
\[ f(3) = 3 \]
2. **Evaluate \( h(f(x)) \) at \( x = 3 \):**
Since \( f(3) = 3 \), now substitute this value into the function \( h(x) \).
\[ h(f(3)) = h(3) \]
\[ h(3) = 6 \cdot 3 + 7 \]
\[ h(3) = 18 + 7 \]
\[ h(3) = 25 \]
Therefore, \( h(f(3)) \) is 25.
### Final Answer:
\[ h(f(3)) \text{ is } 25 \]
[ ] Undefined
![**Problem:**
Find the difference quotient and simplify.
**Function:**
\[ f(x) = -5x^2 - 3x + 2 \]
**Question:**
The difference quotient of \( f(x) \) is: [____]
**Explanation:**
To find the difference quotient of a function \( f(x) \), use the following formula:
\[ \frac{f(x+h) - f(x)}{h} \]
where \( h \) is a small increment.
**Steps to Solve:**
1. Substitute \( x+h \) into the function to find \( f(x+h) \):
\[ f(x+h) = -5(x+h)^2 - 3(x+h) + 2 \]
2. Expand and simplify \( f(x+h) \):
\[ f(x+h) = -5(x^2 + 2xh + h^2) - 3x - 3h + 2 \]
\[ f(x+h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 2 \]
3. Write the difference quotient formula:
\[ \frac{f(x+h) - f(x)}{h} \]
4. Substitute \( f(x+h) \) and \( f(x) \) into the formula:
\[ \frac{(-5x^2 - 10xh - 5h^2 - 3x - 3h + 2) - (-5x^2 - 3x + 2)}{h} \]
5. Simplify the expression inside the numerator:
\[ \frac{-5x^2 - 10xh - 5h^2 - 3x - 3h + 2 + 5x^2 + 3x - 2}{h} \]
\[ \frac{-10xh - 5h^2 - 3h}{h} \]
6. Factor out \( h \) in the numerator:
\[ \frac{h(-10x - 5h - 3)}{h} \]
7. Cancel the \( h \) in the numerator and denominator:
\[ -10x - 5h - 3 \]
Thus, the simplified difference quotient is:
\[ -10x - 5h -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff47ffb76-5051-461b-8a6a-8f4f8799bf23%2F5171b1a7-0b7d-484a-87c9-e311f6bf2f52%2Fll7vhru_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the difference quotient and simplify.
**Function:**
\[ f(x) = -5x^2 - 3x + 2 \]
**Question:**
The difference quotient of \( f(x) \) is: [____]
**Explanation:**
To find the difference quotient of a function \( f(x) \), use the following formula:
\[ \frac{f(x+h) - f(x)}{h} \]
where \( h \) is a small increment.
**Steps to Solve:**
1. Substitute \( x+h \) into the function to find \( f(x+h) \):
\[ f(x+h) = -5(x+h)^2 - 3(x+h) + 2 \]
2. Expand and simplify \( f(x+h) \):
\[ f(x+h) = -5(x^2 + 2xh + h^2) - 3x - 3h + 2 \]
\[ f(x+h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 2 \]
3. Write the difference quotient formula:
\[ \frac{f(x+h) - f(x)}{h} \]
4. Substitute \( f(x+h) \) and \( f(x) \) into the formula:
\[ \frac{(-5x^2 - 10xh - 5h^2 - 3x - 3h + 2) - (-5x^2 - 3x + 2)}{h} \]
5. Simplify the expression inside the numerator:
\[ \frac{-5x^2 - 10xh - 5h^2 - 3x - 3h + 2 + 5x^2 + 3x - 2}{h} \]
\[ \frac{-10xh - 5h^2 - 3h}{h} \]
6. Factor out \( h \) in the numerator:
\[ \frac{h(-10x - 5h - 3)}{h} \]
7. Cancel the \( h \) in the numerator and denominator:
\[ -10x - 5h - 3 \]
Thus, the simplified difference quotient is:
\[ -10x - 5h -
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