Question

Given `g(x)=5/(x−3)`, evaluate and simplify: `(g(5+h)−g(5))/h =`?

Answer 1

`- frac(5)(2h + 4)`

Explanation:

We have: `g(x) = frac(5)(x - 3)`

In order to evaluate `frac(g(5 + h) - g(5))(h)`, we simply substitute `5 + h` and `5` in place of `x` in `g(x)`:

`Rightarrow frac(g(5 + h) - g(5))(h) = frac(frac(5)((5 + h) - 3) - frac(5)((5) - 3))(h)`

`Rightarrow frac(g(5 + h) - g(5))(h) = frac(frac(5)(h + 2) - frac(5)(2))(h)`

`Rightarrow frac(g(5 + h) - g(5))(h) = frac(frac(5 cdot 2 - 5 cdot (h + 2))(2 cdot (h + 2)))(h)`

`Rightarrow frac(g(5 + h) - g(5))(h) = frac(frac(10 - 5h - 10)(2 cdot (h + 2)))(h)`

`Rightarrow frac(g(5 + h) - g(5))(h) = frac(- 5h)(2 cdot (h + 2)) cdot frac(1)(h)`

`therefore frac(g(5 + h) - g(5))(h) = - frac(5)(2h + 4)`

Answer 2

`-5/(2(h+2))`

Explanation:

`"evaluating each term separately"`

`g(5+h)=5/(5+h-3)=5/(2+h)`

`f(5)=5/(5-3)=5/2`

`rArr(g(5+h)-g(5))`

`=5/(2+h)-5/2`

`=(10-5(2+h))/(2(2+h))=(-5h)/(2(2+h))`

`rArr(g(5+h)-g(5))/h`

`=(-5cancel(h))/(cancel(h)2(2+h))=-5/(2(2+h))`