Question

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

Answer 1

a)
`(f(x+h)-f(x))/(h)=(-5)/((x+h-3)(x-3))`

b)
`(f(h-6)-f(-6))/(h)=(5)/(9h-81)`

Explanation:

The given function is
`f(x)=(5)/(x-3)`

`f(x+h)=(5)/(x+h-3)`

`(f(x+h)-f(x))/(h)=((5)/(x+h-3)-(5)/(x-3))/(h)`

`(f(x+h)-f(x))/(h)=(5(x-3)-5(x+h-3))/(h(x+h-3)(x-3))`

`(f(x+h)-f(x))/(h)=(5x-15-5x-5h+15)/(h(x+h-3)(x-3))`

`(f(x+h)-f(x))/(h)=(-5h)/(h(x+h-3)(x-3))`

`(f(x+h)-f(x))/(h)=(-5)/((x+h-3)(x-3))`

When x=-6

`(f(h-6)-f(-6))/(h)=(-5)/((-6+h-3)(-6-3))`

`(f(h-6)-f(-6))/(h)=(-5)/((h-9)(-9))`

`(f(h-6)-f(-6))/(h)=(5)/(9h-81)`