Question

Given the function
g(x) = 9 − x^2

Answer 1

-2x-h

Explanation:

With this problem, we are trying to find the difference quotient, which is what that
`(g(x+h)-g(x))/(h)` means. In order to evaluate that, we need to know what g(x+h) is.

We know that
`g(x)=9-x^2`. For g(x), we know that x is the input the function.

In g(x+h), x+h is the input instead of x.

So for this new function, we are going to plug in (x+h) from where we see x in the original function g(x)

Instead of g(x) = 9-x^2, it is going to be
`g(x+h)=9-(x+h)^2`

See how we replaced x with x+h for the new function g(x+h)?

Now that we know what g(x+h) is, we can go ahead and evaluate the difference quotient.

Remember the thing we are evaluating is
`(g(x+h)-g(x))/(h)`.

We know what g(x+h) and g(x), so instead of g(x+h)-g(x), I am going to put what g(x+h) minus what g(x) is equal to.

So it will look like this now:

`(9-(x+h)^2-(9-x^2))/(h)`

Be really careful with what you are subtracting. Since you are subtracting g(x), you have to put parentheses around what g(x) is equal to.

We can go ahead and simplify this:

`9-(x+h)^2` is the same as
`9-(x^2+2xh+h^2)`

I got
`(x^2+2xh+h^2)` this by using the perfect square trinominal pattern, where
`(a+b)^2 = a^2+2ab+b^2`

With our example, I squared the x, multiplied the x and h by 2, and squared the h.

We can also distribute the negative one to everything inside the parethenses, which will look like:
`9-x^2-2xh-h^2`

Now we have:

`(9-x^2+2xh-h^2-(9-x^2))/(h)`

I am also going to distribute the negative to both the 9 and -x^2,

so instead of
`-(9-x^2)`, it will be
`-9+x^2`

So now we have:

`(9-x^2-2xh-h^2-9+x^2)/(h)`

It looks like we can do some canceling out here.

Let's see, there is a 9 and -9, so 9-9=0

Theres also a -x^2 and x^2, so -x^2+x^2=0

So now we have:

`(-2xh-h^2)/(h)`

Look at the terms in the numerator. The terms are -2xh and -h^2 See how they both have an h. We can factor out an h. So intead of
`-2xh-h^2`, we have
`h(-2x-h)` after factoring out the h.

Now we have:
`(h(-2x-h))/(h)`

We factored out the h so that we can cancel out the h's.

After doing so, we are left with
`-2x-h`, which is our final answer

I am really sorry that this is a long explanation, but I just wanted to go over every step so that it could make sense

If my explanation does not make sense, I am rly rly sorry, but I hope this helps!