Sure! Let's find the difference quotient [tex]\(\frac{f(x + h) - f(x)}{h}\)[/tex] for each given function step-by-step.
### Function 17: [tex]\( f(x) = 9x + 3 \)[/tex]
1. Calculate [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = 9(x + h) + 3
\][/tex]
Expand:
[tex]\[
f(x + h) = 9x + 9h + 3
\][/tex]
2. Form the difference quotient:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{(9x + 9h + 3) - (9x + 3)}{h}
\][/tex]
3. Simplify the expression:
[tex]\[
\frac{(9x + 9h + 3) - 9x - 3}{h} = \frac{9h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{9h}{h} = 9
\][/tex]
The difference quotient for [tex]\( f(x) = 9x + 3 \)[/tex] is:
[tex]\[
\frac{f(x + h) - f(x)}{h} = 9
\][/tex]
### Function 18: [tex]\( f(x) = 5 - 2x \)[/tex]
1. Calculate [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = 5 - 2(x + h)
\][/tex]
Expand:
[tex]\[
f(x + h) = 5 - 2x - 2h
\][/tex]
2. Form the difference quotient:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{(5 - 2x - 2h) - (5 - 2x)}{h}
\][/tex]
3. Simplify the expression:
[tex]\[
\frac{(5 - 2x - 2h) - 5 + 2x}{h} = \frac{-2h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{-2h}{h} = -2
\][/tex]
The difference quotient for [tex]\( f(x) = 5 - 2x \)[/tex] is:
[tex]\[
\frac{f(x + h) - f(x)}{h} = -2
\][/tex]
In summary:
For [tex]\( f(x) = 9x + 3 \)[/tex], the difference quotient is [tex]\( 9 \)[/tex].
For [tex]\( f(x) = 5 - 2x \)[/tex], the difference quotient is [tex]\( -2 \)[/tex].
