Hi Janice,

I want to start with two other functions, p(x) = tan(x) and q(x) = √x. I want to start with these functions since they are buttons on your calculator.

On your calculator press 23, then the tan button and then the square root button. What did you see? After pressing the tan button you saw 0.4245 ( I am only going to use 4 decimal places). Then when you pressed the square root button you saw 0.6515. So

tan(23^o ) = 0.4245 and √0.4245 = 0.6515.

another way to say this is

√tan(23^o ) = 0.6515 or q(p(x)) = 0.6515

This is the function qop(x). First the function p does its thing to x and returns some value (here the value is 0.4245) and then q does its thing to whatever p gives it.

Now lets look at your functions. A description of how g works is that g takes whatever number you give it, multiplies that number by -4 and then adds 3. The function f takes whatever number you give it, squares that number and then multiplies by 2. So, for example if you give 2 to g it returns

g(2) = 3 - 4 2 = -5

If you then give the -5 to f it returns

f(-5) = 2 (-5)² = 50

Thus f(g(2)) = g(-5) = 50. This is fog(2).

So what happens if you give x to g? The function g returns

g(x) = 3 - 4 x.

Now give 3 - 4x to f. The function f returns

f(3 - 4 x) = 2 (3 - 4 x)²

That is

f(g(x)) = 2 (3 - 4 x)² . This is the function fog(x), that is

fog(x) = 2 (3 - 4 x)²

I hope this helps,

Penny