To find the difference quotient of the function [tex]\( f(x) = \frac{13 - 5x}{4} \)[/tex] for all nonzero values of [tex]\( h \)[/tex], we will follow these steps:
1. Write the difference quotient formula:
The difference quotient [tex]\( \frac{f(x+h) - f(x)}{h} \)[/tex] measures the average rate of change of the function [tex]\( f(x) \)[/tex] over an interval of width [tex]\( h \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] and [tex]\( f(x+h) \)[/tex] into the difference quotient:
Given [tex]\( f(x) = \frac{13 - 5x}{4} \)[/tex], we first need to find [tex]\( f(x + h) \)[/tex].
[tex]\[
f(x + h) = \frac{13 - 5(x + h)}{4}
\][/tex]
3. Simplify [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = \frac{13 - 5x - 5h}{4}
\][/tex]
4. Form the difference quotient using [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{13 - 5x - 5h}{4} - \frac{13 - 5x}{4}}{h}
\][/tex]
5. Combine the fractions in the numerator:
[tex]\[
\frac{\frac{13 - 5x - 5h}{4} - \frac{13 - 5x}{4}}{h} = \frac{\frac{(13 - 5x - 5h) - (13 - 5x)}{4}}{h} = \frac{\frac{13 - 5x - 5h - 13 + 5x}{4}}{h}
\][/tex]
6. Simplify the numerator:
[tex]\[
\frac{\frac{13 - 5x - 5h - 13 + 5x}{4}}{h} = \frac{\frac{-5h}{4}}{h}
\][/tex]
7. Divide by [tex]\( h \)[/tex]:
[tex]\[
\frac{\frac{-5h}{4}}{h} = \frac{-5h}{4h} = \frac{-5}{4}
\][/tex]
So, the difference quotient simplifies to [tex]\( -\frac{5}{4} \)[/tex].
Therefore, the correct expression representing the difference quotient of the function [tex]\( f(x)=\frac{13-5x}{4} \)[/tex] for all nonzero values of [tex]\( h \)[/tex] is:
[tex]\[
-\frac{5}{4}
\][/tex]
Thus, the correct answer is:
[tex]\[
-\frac{5}{4}
\][/tex]
