Help on inverse questions, Do not get how to solve

Question

Using the functions f(x) = 2x + 7, g(x) = x2 - 4, and h(x) = 5x. Find all of the following. Make sure your answer is in simplest form.

a. 4h(x)

b. f(x) - g(x)

c. f(g(x))

d. g(x)h(x)

e. g(x) / f(x)

Answer 1

a.) 4*h(x) = 4*(5x) = 20x

 

b.) f(x) - g(x) = (2x+7) - (x²-4) = 2x-x²+11 

 

Now we can simplify this by factoring. Let's rearrange for clarity. We have, -x²+2x+11

Now what multiplies to -11 and adds to 2? Nothing obvious does this so perhaps not as easy as we thought.

You would have to use the quadratic formula to solve this. Let me know if you need help with this.

 

c.) This means you will insert the g(x) function everywhere there is an x in the f(x) function.

Thus,

 

f(g(x)) = 2(x²-4)+7 = 2x²-8+7 = 2x²-1

 

d.) g(x)*h(x) = (x²-4)*(5x) = 5x³-20x

I believe this is the form we get as the answer because if you simplify you will see that you get the original g(x)*h(x)

 

e.) 

 g(x)       x²-4         (x-2)(x+2)

------ = --------- =  --------------

 f(x)       2x+7             2x+7

 

Hope this helps!!