THe notation $\displaystyle f(g(x))$ is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is $\displaystyle 2-2x+x^2$. We will take each x and substitute in the value of g(x), which is 2x-1.

$\displaystyle f(g(x))=f(2x-1)$

$\displaystyle =2-2(2x-1)+(2x-1)^2$

$\displaystyle =2-2(2x-1)+(2x-1)^2$

We will now distribute the -2 to the 2x - 1.

$\displaystyle 2-4x+2+(2x-1)^2$

We must FOIL the $\displaystyle (2x-1)^2$ term, because $\displaystyle (2x-1)^2=(2x-1)(2x-1)$. 

$\displaystyle 2-4x+2+(2x-1)^2=2-4x+2+(2x-1)(2x-1)$

$\displaystyle =2-4x+2+4x^2-2x-2x+1$

Now we collect like terms. Combine the terms with just an x.

$\displaystyle 2+2+4x^2+1-8x$

Combine constants.

$\displaystyle 4x^2-8x+5$

The answer is $\displaystyle 4x^2-8x+5$.

Plug 16 in for $\displaystyle f(x)$.  $\displaystyle 16=x^2-9$

Add 9 to both sides.  $\displaystyle x^2=25$

Take the square root of both sides. $\displaystyle \sqrt{25}$=$\displaystyle \sqrt{x^{2}}$

Final answer is $\displaystyle x=$ $\displaystyle \pm5$

$\displaystyle \small f(g(3))$

This expression is the same as saying "take the answer of $\displaystyle \small g(3)$ and plug it into $\displaystyle \small f(x)$."

First, we need to find $\displaystyle \small g(3)$. We do this by plugging $\displaystyle \small 3$ in for $\displaystyle \small x$ in $\displaystyle \small g(x)$.

$\displaystyle g(x)=x^2$

$\displaystyle g(3)=3^2=9$

Now we take this answer and plug it into $\displaystyle \small f(x)$.

$\displaystyle f(g(3))=f(9)$

We can find the value of $\displaystyle \small f(9)$ by replacing $\displaystyle \small x$ with $\displaystyle \small 9$.

$\displaystyle f(x)=6x-4$

$\displaystyle f(9)=6(9)-4=50$

This is our final answer.

Total cost of the cab ride is going to equal the base fare ($4.50) plus an additional 10 cents per mile. This means the ride will always start off at $4.50. As the cab drives, the cost will increase by $0.10 each mile. This is represented as $0.10 times the number of miles. Therefore the total cost is:

$\displaystyle f(x) = 4.5 + 0.1x$

The pre-question text provides us with all of the information required to complete this problem. 

We know that the total length of the walls is to be $\displaystyle 150$ ft.

We also know that we have a total of $\displaystyle 2 y$ walls and $\displaystyle 4x$ walls.

With this, we can set up an equation and solve for $\displaystyle y$.

Our equation will be with sum of all the walls set equal to the total length of the wall...

$\displaystyle 2y + 4x = 150$

Remeber, we want $\displaystyle y$ in terms of $\displaystyle x$, which means our equation should look like

$\displaystyle y=$ something

$\displaystyle 2y + 4x = 150$  Subtract $\displaystyle 4x$ on both sides

$\displaystyle 2y = 150 - 4x$  Divide by $\displaystyle 2$ on both sides

$\displaystyle 2y/2 = (150-4x)/2$   Simplify

$\displaystyle y = 75 - 2x$     Answer!!!

The function is written in slope-intercept form, which means:

$\displaystyle y = mx+b$

where:

$\displaystyle m$= slope

$\displaystyle x$= x value

$\displaystyle b$= y-intercept

Therefore, the slope is $\displaystyle {\color{Red} -5}$

$\displaystyle f(x)= 29.99+12.99x$

The flat rate of 29.99 does not change depending on months of service.  It is $29.99 no matter how long services are in use.  The monthy fee is directly related to the number of months the services are in use.

To evaluate $\displaystyle f(3)$, we just plug in a $\displaystyle 3$ wherever we see an $\displaystyle x$ in the function, so our equation becomes

$\displaystyle f(3)=\frac{3\cdot 4+3}{3^2}$

which is equal to

 $\displaystyle f(3)=\frac{5}{3}$

To find $\displaystyle f(g+h)$, all we do is plug in $\displaystyle (g+h)$ wherever we see an $\displaystyle x$ in the function. We have to be sure we keep the parentheses. In this case, when we plug in $\displaystyle (g+h)$, we get

$\displaystyle cos(5(g+h)^2)e^{g+h}$

Then, when we expand our binomial squared and distribute the $\displaystyle 5$, we get

$\displaystyle cos(5g^2+10gh+5h^2)e^{g+h}$

First, x needs to be plugged into g(x).

Then, the resulting solution needs to be substituted into f(x).

For example,

$\displaystyle g(7) = 49-1 = 48$. 

$\displaystyle f(7) = g(7)-3 = 48-3 = 45$

Since 45 is an odd number, 7 is an x value that gives this result. Because both equations subtract an odd number to get the final result, only an odd number will result in an odd result therefore, none of the other options will give an odd result.