1. 10. Functions One quantity depends on another quantity
# of shirts -> revenue
Age -> Height
Time of day -> Temperature

2. Function – Definition Formal definition: A function f is a rule that
assigns to each element x in a set A exactly
one element, f (x), in a set B. ( f (x) is read as “ f of x ”)
4 Representations of a function:
Verbally – a description in words
Numerically – a table of values
Algebraically – a formula
Visually – a graph

3. Verbal & Numeric Verbal: Function f(x) divides x by 3 and then adds 4.
Table describes the values in A and their
assignments in B.
Numeric: x -2 5 9 -4 1
f(x)

4. Algebraic & Graphic Algebraic: Graphic:
Plot points (x, f(x)) in the two dimensional plane
f(x) x -2 5 9 -4 1
f(x) 42 21 x 2 4 6 8
-21

5. Example:
Evaluate the function at the indicated values.

6. Piecewise Functions
Evaluate the function at the indicated values.

7. Difference Quotient Find f(a), f(a+h), and the difference quotient
for the given function.
Difference quotient:

8. Domain Set A = input => x = independent variable
Recall definition: A function f is a rule that assigns to each
element x in a set A exactly one element, f(x), in a set B.
Set A = input => x = independent variable
Set B = output => y = f(x) = dependent variable
Domain:
Two restrictions on the domain for any function:

9. Example 1
Find the domain for the following function. D:

10. Example 2
Find the domain for the following function. D:

11. 11. Graphs of Functions
Functions in one variable can be represented by a graph.
Each ordered pair (x, f(x)) that makes the equation true is a point on the graph.
Graph function by plotting points and then connecting the points with smooth curves.

12. Example
f(x)
Create a table of points: 6 x -3 -2 -1 1 x -1 1 -6

13. Basic Functions - Linear
f(x) = mx +b
f(x)
f(x) x x

14. Basic Functions - Power
f(x) =
f(x)
f(x) x x

15. Basic Functions - Root
f(x) =
f(x)
f(x) x x

16. Basic Functions - Reciprocal
f(x) =
f(x)
f(x) x x

17. Basic Functions – Absolute Value
f(x) =
f(x) x

18. Vertical Line Test y= Determine if an equation is a function of x:
Draw a vertical line anywhere and cross the graph at most once, then it is a function. y=
f(x)
f(x) x x

19. Piecewise Function Graph-3

20. Domain/Range from Graph
Look at graph to determine domain (inputs) and range (outputs).
f(x)
Domain: 2
Range: -2 2 4 x -2

21. 12. Average Rate of Change
Average Rate of Change (A.R.O.C.): change in the function values over the change in the input values.
For a function, y = f(x), between x = a and x = b, the A.R.O.C is:

22. Example 1
Find the average rate of change between the indicated points.
A.R.O.C. = y
(-16, 6) 9 3 x
-16
-12 -8 -4 4 8 12 16
(-4, -6) -9

23. Example 2
Find the average rate of change between the indicated points.
A.R.O.C. = y
(-16, 6)
(12, 9) 9 3 x
-16
-12 -8 -4 4 8 12 16 -9

24. Example 3 Find the average rate of change of the function between
the values of the variable.
A.R.O.C.

25. Example 4 Find the average rate of change of the function between
the values of the variable.
A.R.O.C.

26. Example 5
A man is running around a track 200 m in circumference. With the use of a stopwatch, his time is recorded at the
end of each lap, seen in the table below.
What was the man’s speed between 66s and 209s? Round answer to the nearest hundredth.
Time (s)
Distance (m) 32
200 66
400
104
600
153
800
209
1000
270
1200
341
1400
419
1600
Calculate the man’s speed for each lap.
Please round answer to the nearest hundredth.
Lap 1:
Lap 2:

27. 13. Combining Functions Take simple functions and combine for more
complicated ones
Arithmetic - add, subtract, multiply, divide
Composition – evaluate one function inside another

28. Arithmetic Combinations
Given the functions:
Domain:
Domain:
Domain:
Domain:

29. Composition

30. Example 1
Find the composition function:
Domain:

31. Example 2
Find the composition function:
Domain:

32. Example 3

33. Example 4
Use the graphs to evaluate: 2 -3 -1 4 -2

34. Application
An airplane is flying 300 mi/hr at an altitude of 2 miles. At t = 0, the plane passes directly over a radar station.
Express s as a function of d. d 2 s
Express d as a function of t.
Express s as a function of t.

35. 14. Transformations Vertical Shifts
f(x) + c: Add to the function => shift up
f(x) - c: Subtract from the function => shift down

36. Horizontal Shifts f(x+c): Add to the variable => shift back(left)
f(x – c): Subtract from the variable => shift forward(right)

37. Example: Shifts
Graph the function:

38. y-axis reflection
f(-x): Negate the variable => reflect in the y-axis

39. x-axis reflection
-f(x): Negate the function => reflect in the x-axis

40. Example: Shift and reflection
Graph the function:

41. Vertical Stretch & Shrink
cf(x): multiply the function by c > 1=> vertical stretch
(8,2)
(1,1)
cf(x): multiply the function by 0 < c < 1 => vertical shrink
(2,4)

42. Even & Odd Functions Even function(y-axis symmetry): if f(-x) = f(x)
Odd function(origin symmetry): if f(-x) = -f(x)
Determine whether the following are even, odd, or neither.

43. 15. Inverse Functions
One-to-one functions: a function is one-to-one if every input is associated with one output and each output is associated with only one input.
Horizontal Line Test – a function is one-to-one if and only if no horizontal line intersects the graph more than once.

44. Inverses Every one-to-one function, f(x), has an associated
Function called an inverse function, f -1(x).
The inverse function reverses what the function does.
Its input is another function’s output.
Its output is another function’s input. 3
.77 5 A B 4 -2 4

45. Example 1

46. Finding Inverses Graphically
Inverses swap x and y coordinates. 2 -3 -1 4 -2

47. Finding Inverses Algebraically
Three step process:
Write the equation y = f (x).
Solve the equation for x in terms of y.
Swap the x and y variables.
The resulting equation is y = f -1(x).

48. Example 2
Find the inverse function for:

49. Example 3
Find the inverse function for:

50. Inverse Property