Question
Given the function \( f(x)=4 x+8 \), evaluate and simplify the expressions below. See special instructions on how to enter your answers. \( f(a)= \) \( f(a+h)=\square \) \( \frac{f(a+h)-f(a)}{h}=\square \) Instructions: Simplify answers as much as possible. Expressions such as \( 4(x+2) \) and \( (x+5)^{2} \) should be expanded. Also collect like terms, so \( 3 x+x \) should be written as \( 4 x \).
Ask by Howell Bates. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
f(a) = 4a + 8
\]
\[
f(a+h) = 4a + 4h + 8
\]
\[
\frac{f(a+h) - f(a)}{h} = 4
\]
Solution
Let's evaluate and simplify each expression step by step for the function \( f(x) = 4x + 8 \).
1. **Evaluate \( f(a) \):**
\[
f(a) = 4a + 8
\]
2. **Evaluate \( f(a+h) \):**
\[
f(a+h) = 4(a + h) + 8 = 4a + 4h + 8
\]
3. **Evaluate \( \frac{f(a+h) - f(a)}{h} \):**
\[
\frac{f(a+h) - f(a)}{h} = \frac{(4a + 4h + 8) - (4a + 8)}{h} = \frac{4h}{h} = 4
\]
**Final Answers:**
\[
f(a) = 4a + 8
\]
\[
f(a+h) = 4a + 4h + 8
\]
\[
\frac{f(a+h) - f(a)}{h} = 4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Let's evaluate the function step-by-step. 1. **Evaluate \( f(a) \):** \[ f(a) = 4a + 8 \] 2. **Evaluate \( f(a+h) \):** \[ f(a+h) = 4(a+h) + 8 = 4a + 4h + 8 \] 3. **Calculate \( \frac{f(a+h) - f(a)}{h} \):** \[ f(a+h) - f(a) = (4a + 4h + 8) - (4a + 8) = 4h \] Thus, we divide by \( h \): \[ \frac{f(a+h) - f(a)}{h} = \frac{4h}{h} = 4 \] So the simplified results are: - \( f(a) = 4a + 8 \) - \( f(a+h) = 4a + 4h + 8 \) - \( \frac{f(a+h)-f(a)}{h} = 4 \)
