Function Operations

True or false: It is always true that  $\left(f+g\right)\left(x\right)\ =\ \left(g+f\right)\left(x\right)$ . 

If  $f\left(x\right)\ =\ -x^2+3$  and  $g\left(x\right)\ =\ x^3+2x^2-2$ , then  $\left(f-g\right)\left(x\right)\ =\ ...$   

Given $f\left(x\right)\ =\ 3x^2+x+4$ , find two non-zero functions  $g\left(x\right)$  and  $h\left(x\right)$  such that  $f\left(x\right)\ =\ \left(g-h\right)\left(x\right).$  

Let $f\left(x\right)\ =\ 2x-2x^2$ . Select all of the following ways  that  $f\left(x\right)$  can be written as a product,  $f\left(x\right)\ =\ \left(g\cdot h\right)\left(x\right)$ . 

Let $f\left(x\right)\ =\ x^2-25$ and  $g\left(x\right)\ =\ x^2-7x+10$ . Find the quotient  $\left(\frac{g}{f}\right)\left(x\right)$  in simplified form.

Make sure you enter your answer with all necessary parentheses. For example, you could enter  $\frac{2}{x+1}$  as 2/(x+1).  
  

(a)  

Select all that apply: the division of functions often results in...

Let $f\left(x\right)\ =\ e^x$ and  $g\left(x\right)\ =\ x^2$ . Then  $f\left(g\left(x\right)\right)\ =\ ...$   

Let  $f\left(x\right)\ =\ \sin\left(x\right)$ ,  $g\left(x\right)\ =\ 3x^2-1$ , and  $h\left(x\right)\ =\ 2x$ .

Find  $f\left(g\left(h\left(x\right)\right)\right)$ .  

Consider the composition of four functions $f\left(x\right)\ =\ \cos^2\left(4e^x-1\right)+6$ . Which of the following choices lists the four functions from innermost to outermost?

(hint: think about order of operations!)

If  $f\left(x\right)\ =\ x+h$  and  $g\left(x\right)\ =\ x^2+\frac{3}{x}$ , then  $g\left(f\left(x\right)\right)\ =\ ...$