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

Answer:

$f(a)=8a+5\\\\f(x+h)=8x+8h+5\\\\\frac{f(x+h)-f(x)}{h}=8$

Step-by-step 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:

$\frac{(8x+8h+5)-(8x+5)}{h}$

Now let's distribute the negative sign to get:

$\frac{8x+8h+5-8x-5}{h}$

Now let's group like terms:

$\frac{(8x-8x)+(5-5)+8h}{h}$

Now let's simplify the grouped terms

$\frac{8h}{h}$

Now let's divide by h to get:

$8$

And this is our answer!