To solve for the difference quotient [tex]\(\frac{f(x + h) - f(x)}{h}\)[/tex] for the function [tex]\(f(x) = 23x - 12\)[/tex], we need to follow these steps:
1. Define [tex]\(f(x)\)[/tex] and [tex]\(f(x + h)\)[/tex]:
[tex]\[
f(x) = 23x - 12
\][/tex]
[tex]\[
f(x + h) = 23(x + h) - 12
\][/tex]
2. Expand and simplify [tex]\(f(x + h)\)[/tex]:
[tex]\[
f(x + h) = 23(x + h) - 12 = 23x + 23h - 12
\][/tex]
3. Find the difference [tex]\(f(x + h) - f(x)\)[/tex]:
[tex]\[
f(x + h) - f(x) = (23x + 23h - 12) - (23x - 12)
\][/tex]
[tex]\[
= 23x + 23h - 12 - 23x + 12
\][/tex]
[tex]\[
= 23h
\][/tex]
4. Divide the result by [tex]\(h\)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{23h}{h}
\][/tex]
5. Simplify the division:
[tex]\[
\frac{23h}{h} = 23
\][/tex]
Thus, the difference quotient [tex]\(\frac{f(x + h) - f(x)}{h}\)[/tex] for the function [tex]\(f(x) = 23x - 12\)[/tex] is 23.
Considering the provided options:
A) 23
B) 11
C) 23x
D) -23
The correct answer is:
A) 23
