Question

Given the function f(x) = 8x +5, evaluate and simplify the expressions below. See special instructions on how to enter your answers. Look at photo for the expressions and special instructions

Answer 1

`f(a)=8a+5\\\\f(x+h)=8x+8h+5\\\\(f(x+h)-f(x))/(h)=8`

Explanation:

Substitution:

When we're asked to write what f(x+h) or f(a) is equal to, all we really have to do is substitution. The original function is provided as
`f(x)=8x+5`, and all that "x" represents is input, now we can manipulate what that input is by simply substituting in new input. So if we want to write f(x+h), we substitute in "x+h" for "x".

f(a) = ?

For this problem, we just do what was stated about and replace all of the x's, with a's, meaning:
`8x+5\to 8a+5`.

f(x+h) = ?

We essentially do the same thing here, although we will have to simplify more, but the first step is as follows:
`8x+5\to 8(x+h)+5`, now from here we want to expand out the
`8(x+h)` by using the distributive property, so it becomes:
`8x+8h` and plugging this into our equation we get:
`8x+8h+5`

[f(x+h)-f(x)]/h = ?

For this last question, I think it's also important to note that this is the formula to calculate the slope between point "x" and the point "h units away" or "x+h".

We already know what f(x+h) is and what f(x) is, so let's plug those into the equation:

`((8x+8h+5)-(8x+5))/(h)`

Now let's distribute the negative sign to get:

`(8x+8h+5-8x-5)/(h)`

Now let's group like terms:

`((8x-8x)+(5-5)+8h)/(h)`

Now let's simplify the grouped terms

`(8h)/(h)`

Now let's divide by h to get:

`8`

And this is our answer!