The document provides steps for finding the inverse of the function f(x) = 12x-4/x-7. It works through rearranging the equation by switching x and y values, distributing terms, and factoring to arrive at the inverse function f-1(x)=7x-4/(x-12). A second example shows how to find the maximum area of a playground given its perimeter of 18600 feet by setting up equations for perimeter and area and solving.

1. A L E X A N D R I A
B R O W N A N D
S A M A N T H A
P E R K I N S .

2. Inverses
f(x) = 12x-4/x-7
Step 1: Switch f(x) to x= and switch all x’s in equation to y’s. So x=12y-4÷y-7
Step 2: multiply the denominator (x-7) to both sides so x(x-7)=12y-4(y-7)÷(y-7)… (x-7)
divided by (x-7) cancels out to 1. So that being said you are left with x(y-7)=12y-4.
Step 3: Distribute x(y-7) so… x times y= xy, and x times 7= 7x
So… xy-7x=12y-4.
Step 4: Get y’s on one side and x’s on the other. Do this by subtracting 7x from both
sides and subtracting 12y from both sides.
So the new equation… xy-12y=7x-4
Step 5: Factor out y from xy-12y. So… y(x-12)=7x-4
Step 6: Divide out (x-12) to finalize the inverse function!
So… y=7x-4÷(x-12)
Step 7: Change “y=“ f-1(x)=7x-4÷x-12

3. Construction Worker Yoda
Farmer Yoda is building a new playground for Chewbacca. The perimeter of this is
18600 feet. Keeping that in mind find the maximum area of this playground.
Step 1: Consider the perimeter and area equations, so Perimeter=2x+2y and the
equation for Area=x•y
Step 2: Insert the perimeter of the playground into the perimeter equation.
So… 18600=2x+2y
Step 3: Solve for y. Do this by dividing by 2 on both sides. So… 18600÷2=2x+2y÷2
After calculating that you get 9300=x+y. Then subtract x from both sides leaving you
with the equation y=9300-x.
Step 4: Insert the quantity (9300-x) into the area equation (A=x•y)
So… A=x(9300-x)
Step 5: Distribute x(9300-x) So… x2 + 9300x
Step 6: -b÷2a to find the x value so the -9300÷2=-4650 and x=4650
Step7: Insert 4650 for x in x(9300-x) so 4650(9300-4650) this finds the maximum area.
Step 8: Maximum area = 21,622,500 ft2

4. Graphing Polynomials
Things to keep in mind:
There are four different ways a polynomial graph could look.
Odd - Odd + Even + Even –
-13x3 + 9x2 +9x +17 7x3 +6x2 + 3x-20 6x4 -9X3+x2 +3x-3 -3x2 +2x+2
You decide the picture of the graph by the first set.
For example: 4x3 would be Odd +
positive value odd exponent

5. What else?
Well..
If the factor pair has a single exponent, meaning no
physical exponent at all, when drawn, the line goes
straight through the x-intercept.
Example: (x-7)
no exponent
7
For odd powered factor pairs, you curve through the x-intercept.
Example: (x-5)5
five is odd
* MAKE THE CURVE OBVIOUS 5
If a factor pair has an even number for an exponent, when drawn,
the line will bounce on the x-intercept.
Example: (x-2)2
two is even
2
If there is no factor pair, and its just x,
such as x9 or x4, the x intercept would be
zero, but the exponent rules still apply
for when you draw it out!

6. So can we just graph it now or
Task: graph the polynomial x4 (x+8) (x-1)3
What will it look like? Look at the first intercept that’s given.
x4
positive even We know that Even+ graphs look like….
Our graph will have this shape. Now look at all the x-intercepts.
x4 (x+8) (x-1) 3
even gets a bounce odd gets a curve
nothing gets straight
Remember that you’re plotting what x would be to make the factors = 0.
So to plot, you have..
x= 0 (curve) x= -8 (straight) x= 1 (curve)
-8 0 1
Plot your points and connect!
straight bounce curve

7. Long division with Exponents

8. 3 7 12
3
8
1
2
Answer Key
g(7)=3
f(3)=12
h(12)=8
Step 1: look at g-graph, and
search where that graph has an x-
value of 7. Then search for the y-
value where x is 7, on this graph it
is 3. So g(7)=3
Step 2: Now it’s time to look at
the f-graph. Since the answer to
g(7) was 3, now you must search
where the f-graph=3. So f(3)=12
Step 3: Now that you have found
f(3) at 12, now you must find
h(12).. To do this, search for the
h-graph. Now look where the h-
graph has an x-value of 12. Once
you have found that you search
for the y-value of the graph where
the x-value is 12. So h(12)=8.
So… h(f(g(x)))=8!
g
hf