I'm having trouble with these problems

Question

Given that f(x)=x2−6xf(x)=x2-6x and g(x)=x+3g(x)=x+3, find the unsimplified AND simplified versions of the following compositions:

a) (f∘g)(x)(f∘g)(x)=    unsimplified

   (f∘g)(x)(f∘g)(x)=    simplified

b) (f∘f)(x)(f∘f)(x)=    unsimplified

   (f∘f)(x)(f∘f)(x)=    simplified

Answer 1

Hi Tra'von, I'm going to answer what I believe to be the questions here. There was an issue with what you pasted, where everything is pasted twice.

f(x) = x²-6x, and g(x) = x+3

a) (f o g)(x) is the same as saying "f of g of x", or f(g(x)). So we do what's in the parenthesis first, substituting g(x) = x+3.

f(g(x))

= f(x+3)

= (x+3)² - 6(x+3). [This is the unsimplified version]

Now we simplify that by foiling the square and collecting like terms together.

(x+3)² - 6(x+3)

= x²+6x+9 - 6x - 18

= x² - 9 [simplified]

b) (f o f)(x) = f(f(x)). Same thing here - we substitute f(x) = x²+6x into the parenthesis.

f(f(x))

= f(x²+6x)

= (x²+6x)² + 6(x²+6x) [Unsimplified]

Now we again simplify by foiling out and collecting like terms.

(x²+6x)² + 6(x²+6x)

= x⁴+12x³+36x² + 6x²+36x

= x⁴+12x³+42x²+36x [Simplified]