Little Lynch
11/21/2023 · Senior High School
Determine the difference quotient \( \frac{f(x+h)-f(x)}{h} \) \( f(x)=4 x-8 \)
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Tutor-Verified Answer
Step-by-step Solution
To determine the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = 4x - 8 \), we first need to calculate \( f(x+h) \).
1. **Calculate \( f(x+h) \)**:
\[
f(x+h) = 4(x+h) - 8
\]
Expanding this, we get:
\[
f(x+h) = 4x + 4h - 8
\]
2. **Calculate \( f(x) \)**:
We already have:
\[
f(x) = 4x - 8
\]
3. **Substitute into the difference quotient**:
Now we substitute \( f(x+h) \) and \( f(x) \) into the difference quotient:
\[
\frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h - 8) - (4x - 8)}{h}
\]
4. **Simplify the numerator**:
Simplifying the expression in the numerator:
\[
(4x + 4h - 8) - (4x - 8) = 4x + 4h - 8 - 4x + 8 = 4h
\]
5. **Complete the difference quotient**:
Now we have:
\[
\frac{4h}{h}
\]
As long as \( h \neq 0 \), we can simplify this to:
\[
4
\]
Thus, the difference quotient \( \frac{f(x+h) - f(x)}{h} \) is \( \boxed{4} \).
Quick Answer
The difference quotient is 4.
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