Question

Given f(x) = 5/(x-3), simplify f(x+h)-f(x)/h, h does not equal 0 when x = -6

Answer 1

Answer:

a) $\frac{f(x+h)-f(x)}{h}=\frac{-5}{(x+h-3)(x-3)}$

b) $\frac{f(h-6)-f(-6)}{h}=\frac{5}{9h-81}$

Step-by-step explanation:

The given function is $f(x)=\frac{5}{x-3}$

$f(x+h)=\frac{5}{x+h-3}$

$\frac{f(x+h)-f(x)}{h}=\frac{\frac{5}{x+h-3}-\frac{5}{x-3}}{h}$

$\frac{f(x+h)-f(x)}{h}=\frac{5(x-3)-5(x+h-3)}{h(x+h-3)(x-3)}$

$\frac{f(x+h)-f(x)}{h}=\frac{5x-15-5x-5h+15}{h(x+h-3)(x-3)}$

$\frac{f(x+h)-f(x)}{h}=\frac{-5h}{h(x+h-3)(x-3)}$

$\frac{f(x+h)-f(x)}{h}=\frac{-5}{(x+h-3)(x-3)}$

When x=-6

$\frac{f(h-6)-f(-6)}{h}=\frac{-5}{(-6+h-3)(-6-3)}$

$\frac{f(h-6)-f(-6)}{h}=\frac{-5}{(h-9)(-9)}$

$\frac{f(h-6)-f(-6)}{h}=\frac{5}{9h-81}$