1. Welcome to the Webinar “Live” Review for Test 3 (MAC 1105)
We will start promptly at 8:00 pm
Everyone will be placed on mute
I am happy that you are able to join us 

2. “Live” Review for Test 3 (MAC 1105)
The following slides present sample problems similar to the test content, including the worked-out solutions.
These are a few problems and do not include all the competencies tested on Test 3.
To be prepared for the test, you must review the material from all the corresponding sections and follow the Study Guide posted on Blackboard!

3. The function whose graph is shown below is a one-to-one function.
True or False
(-3, 3) (3, 3)
Answer: False
Does not pass the horizontal line test. In a 1-1 function different inputs will always result in different outputs. In our example, inputs -3 and 3 yield the same output, 3; The given function is not 1-1.
Remember for a function, one input gives one output, and for a one-to-one function, one output gives one input.

4. The function f(x) = 2x + 5 is a one-to-one function. a. True b. False
y = 2x + 5 is a linear function
Algebraically:
Answer: True
Passes horizontal line test; two different inputs will not yield the same output.
Input -2
1/2 3 10
Output 1 5 6 11 25

5. Find the inverse of f(x) = {(-5, 6), (0, 9), (1/2, -6), (3, 4)}
We must check something first! What?
Is this a one-to-one function?
Your time to work!

6. Find the inverse of f(x) = {(-5, 6), (0, 9), (1/2, -6), (3, 4)}
This is a one-to-one function, since different inputs produce
different outputs.
Finding the inverse: Interchange the x-and-y coordinates.
So, f -1(x) = {(6, -5), (9, 0), (-6, 1/2), (4, 3)}

7. Find the inverse function for: f(x) = x + 25
y = x + 2
Interchange variables:
x = y + 2
Solve for new y: first multiply both sides by 5
(5)x = (y + 2) (5)
5x = y + 2
  y = 5x So, f-1(x)= 5x - 2

8. Find the domain and range of the inverse:
{(-5, 6), (0, 9), (1/2, -6), (3, 4)}
The domain of the inverse is the range of the original function.
 The range of the inverse is the domain of the original function.
So,
Domain of inverse = 6; 9; -6; 4
Range of inverse = -5; 0; 1/2; 3

9. State whether the inverse exists: y = x2 – 5.
This is a parabola; it fails the horizontal line test, thus, it is
not one-to-one.
Attempting to find the “inverse” would result in y =  sqrt(x + 5)
whose graph is
where, with the exception of x = -5, every input would have two
different outputs. Not a function! We would need to restrict the
domain of the given function.

10. If f(x) = 2x – 1 and g(x) = x + 4, find (f – g)(x).
(f – g)(x) = f(x) – g(x)
= (2x – 1) – (x + 4) = 2x – 1 – x – 4 = x – 5
Find (f  g)(x)
(f  g)(x) = f(x)  g(x) = (2x – 1)(x + 4) = 2x2 + 7x – 4
Find (f/g)(x)
(f/g)(x) = f(x)/g(x) = 2x – 1
x + 4
x  –4

11. If f(x) = 2x – 1 and g(x) = x + 4, find (f – g)(3).
(f – g)(x) = f(x) – g(x)
Two approaches:
Find (f – g)(x) and then find (f – g)(3).
(f – g)(x) = (2x – 1) – (x + 4) = 2x – 1 – x – 4 = x – 5
So, (f – g)(3) = 3 – 5 = -2 OR
Find f(3) and g(3) and then subtract.
f(3) = 2(3) – 1 = and g(3) = = 7
So, (f – g)(3) = 5 – 7 = -2

12. Please select your answer.
Determine which represents "g composed with f "
Please select your answer.

13. Example: If f(x) = 2x + 3 and g(x) = x – 8
"g composed with f”
That is, function f will be the input for function g.
Example: If f(x) = 2x and g(x) = x – 8
then “2x + 3” will be the input for function g.
Since g(x) = x – 8
g(f(x)) = (2x + 3) – 8
= 2x – 5

14. If f(x) = -x2 and g(x) = , find (f o g)(x)
(f o g)(x) = f(g(x))
That is, function g will be the input for function f.
So, f(g(x))
= f( )
= - ( )2
= -(x + 2)
= -x − 2

15. If f(x) = -5x and g(x) = x2 + 2, find (g o f)(3).
One approach: Calculate (g o f)(x) = g(f(x)) and then evaluate the resulting function when x = 3
g(f(x)) = g(-5x) = (-5x)2 + 2 = 25x2 + 2  
Thus, (g o f)(3) = 25(3)2 + 2 = 227
Another approach: Since (g o f)(x) = g(f(x)), calculate f(3), then substitute the resulting value into g and evaluate:
f(3) = -5(3) = -15
Then, g(-15) = (-15)2 + 2 = 227

16. Degree?
Highest exponent is 3, therefore degree is 3
Leading term?
-x³
Leading coefficient? -1
Constant term? -8

17. Use the graph to determine the following for the polynomial function:
Even or odd degree?
Positive or negative leading coefficient?
Number of turning points?
Odd degree: One end of the graph points up; the other end points down.
Negative: The right-hand end points down.
Turning points: 2

18. Your time to answer the question!
Even or odd degree? Positive or negative leading coefficient?
Your time to answer the question!

19. Even or odd degree? Positive or negative leading coefficient?
Both ends point down: Even degree and negative leading coefficient

20. Determine the intercepts of the polynomial function
To find y-intercept: Let x = 0; find f(0).
To find x-intercepts: Let y = 0; solve f(x) = 0
y-intercept:
f(0) = (0 – 2)(30 – 1)(0 + 4) = (-2)(-1)(4) = 8
y = 8 or the ordered pair (0, 8)
x-intercepts:
(x – 2)(3x – 1)(x + 4) = 0
x – 2 = x – 1 = x + 4 = 0
x = 2, 1/3, -4 or the ordered pair (2, 0), (1/3, 0), (-4, 0)

21. The graph of f(x) = (x – 1)( x + 3)(x – 3)² will cross the x-axis at
True or False?
The zeros of the function are: -3, 1, 3
1 with multiplicity 1
-3 with multiplicity 1
3 with multiplicity 2
If k is a zero of odd multiplicity, the graph will cross the x-axis
at k.
If k is a zero of even multiplicity, the graph will touch the x-axis
and turn around at k.
So, the answer is False, because x = 3 is a zero with an even
multiplicity.

22. Determine the factored form of the equation of the polynomial
whose graph is shown below.
x(x + 4)(x – 4)
b. -x(x + 4)(x – 4)
c. (x + 4)²(x – 4)²
d. (x + 4)(x – 4)²
First, one end points up, the other end points down, so it is an odd-degree polynomial. (Cannot be “c”)
It crosses the origin, so x = 0 is a real zero. (Cannot be “d”)
Right-hand end points down, so, the leading coefficient is negative.
Answer: b

23. Find the domain of the rational function:
y =
  x² – 49
The denominator cannot equal zero:
x² – 49  0
(x + 7) (x – 7)  0
x  -7 and x  7
or in interval notation (-infinity -7) U (-7, 7) U (7, infinity)

24. Find the vertical asymptote(s) of
x = 2/5, 3
x = -2/5, -3
x = 11/5, -11/5
x = 0
5x² – 17x + 6 = 0
(5x – 2)(x – 3) = 0
5x – 2 = 0 or x – 3 = 0
x = 2/5, 3
Answer is: a
y = x
  x² – 17x + 6

25. Horizontal Asymptotes
Let m = degree of numerator and n = degree of denominator
If m < n horizontal asymptote is line y = 0 (the x-axis)
If m = n horizontal asymptote is ratio of leading coefficients
If m > n no horizontal asymptote
Find the horizontal asymptote, if it exists:
a. y = x² b. y = x c. y = x
  x² – 3x x² – 3x x² – 3x + 6
m = n So, y = 7 = 7 is a horizontal asymptote 1
m > n So, there is no horizontal asymptote
c. m < n So, y = 0 is a horizontal asymptote

26. Find the domain and the x-intercepts:
Domain: x  0 or in interval notation (-infinity, 0) U (0, infinity)
x-intercepts: Solve f(x) = 0,
(x – 5) (x + 6) = 0
So, x-intercepts: 5, -6 or in ordered pair (5, 0),(-6, 0)
NOTE: It is important that you follow the instructions on the test. If asked for ordered pair, do so!
If you are not asked for ordered pair, then just write the values separated by commas (,) or semicolons (;) as specified on the question.

27. The average cost, A, per item of producing x number of a new product-
item is given by A(x) = x
  x
a. Find and interpret A(400).
b. If the average cost per item were $7.50, how many could they
produce?
c. Graph the average cost function using [0, 500, 50] by [0, 100, 10].
a. A(400) = (400) = 6.8  
The average cost per item of producing 400 items is $6.80.
b x = 7.5
  x
(x) x = 7.5 (x)
  x
x = 7.5x
x =  276 items

28. c. Graph the average cost function using [0, 500, 50] by [0, 100, 10].
y = x
  x
Y1 = ( x)/x
Average cost
Number of T-shirts

29. This "live" review session has covered content addressed on test 3, but you need to review chapters 5 and 7
and complete all the problems assigned on the study guide, so that you are well-prepared for the test.

30. Good luck  Do Well!