Question

No. 79. How is the answer -3/x(x+h)?

Answer 1

See explanation.

Explanation:

Number 79 says that `f(x)=3/x`. As defined, this function `f` takes in a value `x`, then returns the quotient of `"3 divided by "x.`

More generally, when we give `f` some value, whatever it is, `f` takes it, divides 3 by it, and gives us back the result.

When `f` takes the input `4`, the output from `f` is `3/4`.
When `f` takes the input `x`, `f` returns `3/x`.
And when `f` is given `x+h`, its output will be `3/(x+h).`

With all this, we now simplify `(f(x+h)-f(x))/h`:

`(color(blue)(f(x+h))-color(green)(f(x)))/h=(color(blue)(3/(x+h))-color(green)(3/x))/(h)`

`color(white)((f(x+h)-f(x))/h)=((3x)/color(red)(x(x+h))-(3(x+h))/color(red)(x(x+h)))/color(orange)h`

`color(white)((f(x+h)-f(x))/h)=(3x-3(x+h))/(color(orange)hcolor(red)(x(x+h)))`

`color(white)((f(x+h)-f(x))/h)=(cancel(3x)-cancel(3x)-3h)/(hx(x+h))`

`color(white)((f(x+h)-f(x))/h)=(-3cancelh)/(cancelhx(x+h))`

`color(white)((f(x+h)-f(x))/h)=(-3)/(x(x+h))`