Let's begin by analyzing the given function [tex]\( f(x) = 9x + 3 \)[/tex].
We need to find the expression for [tex]\( \frac{f(x+h) - f(x)}{h} \)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = 9(x + h) + 3
\][/tex]
Simplify the expression:
[tex]\[
f(x + h) = 9x + 9h + 3
\][/tex]
2. Calculate [tex]\( f(x + h) - f(x) \)[/tex]:
[tex]\[
f(x + h) - f(x) = (9x + 9h + 3) - (9x + 3)
\][/tex]
Simplify the expression by combining like terms:
[tex]\[
f(x + h) - f(x) = 9x + 9h + 3 - 9x - 3
\][/tex]
[tex]\[
f(x + h) - f(x) = 9h
\][/tex]
3. Divide [tex]\( f(x + h) - f(x) \)[/tex] by [tex]\( h \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{9h}{h}
\][/tex]
Simplify the expression by canceling out [tex]\( h \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = 9
\][/tex]
Therefore, the result is:
[tex]\[
\frac{f(x+h) - f(x)}{h} = 9
\][/tex]
So the difference [tex]\( f(x + h) - f(x) \)[/tex] is [tex]\( 9h \)[/tex] and the quotient [tex]\( \frac{f(x + h) - f(x)}{h} \)[/tex] is [tex]\( 9 \)[/tex].
