Let's solve this step-by-step for the given function [tex]\( f(x) = \frac{5}{x} \)[/tex].
### Part a: Calculate [tex]\( f(x + h) \)[/tex]
The expression [tex]\( f(x + h) \)[/tex] simply means substituting [tex]\( x \)[/tex] with [tex]\( x + h \)[/tex] in the function [tex]\( f(x) \)[/tex].
Therefore,
[tex]\[ f(x + h) = \frac{5}{x + h} \][/tex]
### Part b: Calculate [tex]\( f(x + h) - f(x) \)[/tex]
Next, we need to find the difference between [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex]. Using the expressions for [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex], we have:
[tex]\[ f(x + h) - f(x) = \frac{5}{x + h} - \frac{5}{x} \][/tex]
### Part c: Calculate [tex]\(\frac{f(x + h) - f(x)}{h}\)[/tex]
Finally, we need to find the difference quotient, which is the expression from part b divided by [tex]\( h \)[/tex]:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{\left(\frac{5}{x + h} - \frac{5}{x}\right)}{h} \][/tex]
So, summarizing all parts:
a. [tex]\( f(x+h) = \frac{5}{x + h} \)[/tex]
b. [tex]\( f(x+h) - f(x) = \frac{5}{x + h} - \frac{5}{x} \)[/tex]
c. [tex]\(\frac{f(x+h) - f(x)}{h} = \frac{\left(\frac{5}{x + h} - \frac{5}{x}\right)}{h} \)[/tex]
These are the detailed steps for finding [tex]\( f(x+h) \)[/tex], [tex]\( f(x+h) - f(x) \)[/tex], and [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] for the given function [tex]\( f(x) = \frac{5}{x} \)[/tex].
