1. College Algebra: Lesson 1
Functions and Graphs

2. Topics Topics include basics of functions and their graphs
linear functions and slope
transformations of functions
combinations of functions/composite functions
inverse functions

3. Relations A relation is any set of ordered pairs.
{(1,2), (3,4), (5,6)}
The domain comprises the set of first elements.
{1, 3, 5}
The range comprises the set of second elements.
{2, 4, 6}

4. Functions
A function is a relation for which each element in the domain corresponds to exactly one element in the range.
A function can have two different first components with the same second component.
{(1, 2), (3, 2), (4, 1)}
A function cannot have the same first component and different second components.
{(1, 2), (3, 2), (1, 3)}

5. Functions
Functions are usually expressed in terms of equations involving x and y where x is the independent variable and y is the dependent variable.
If an equation defines a function, the value of the function at x, f(x), often replaces y.

6. Example y = 2x + 6 Evaluate for x = 3 f(x) = 2x + 6 Evaluate for x = 3
Equation
Functional Notation
y = 2x + 6
Evaluate for x = 3
y = 2(3) + 6 = 12
f(x) = 2x + 6
Evaluate for x = 3
f(3) = 2(3) + 6 = 12
Example

7. Graphs of functions
The graph of a function is the graph of its ordered pairs.
Not every graph is the graph of a function.

8. Vertical Line Test
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

9. Relation
Function
Example

10. Obtaining Information from Graphs
A closed dot indicates that the graph does not extend beyond this point AND the point belongs to the graph.
An open dot indicates that the graph does not extend beyond this point AND the point does not belong to the graph.
An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.

11. Obtaining Information from Graphs
To find the domain, look for all of the inputs on the x-axis that correspond to points on the graph.
To find the range, look for all of the inputs on the y-axis that correspond to points on the graph.

12. Example Use the graph of y = f(x) to solve this exercise.
Domain and Range
Graph
Use the graph of y = f(x) to solve this exercise.
What is f(4) – f(-3)?
f(4) = 3, f(-3) = -2
f(4) – f(-3) = 5
What are the domain and range of f?
Domain (-5, 6]
Range [-4, 5]
Example

13. Obtaining Information from Graphs
To find the x-intercepts, look for the points at which the graph crosses the x- axis.
The zeros of a function f are the x-values for which f(x) = 0.
The real zeros are the x-intercepts.
To find the y-intercept, look for the points at which the graph crosses the y-axis.
A function can have more than one x- intercept but at most one y-intercept.

14. Example Use the graph of y = f(x) to solve this exercise. Recall that
X- and y-Intercepts
Graph
Use the graph of y = f(x) to solve this exercise.
What are the x-intercepts?
-4, 1, 5
What is the y-intercept? -3
Recall that
A function can have more than one x-intercept but at most one y-intercept.
Example

15. Example Use the graph of y = f(x) to solve this exercise. Recall that
Zeros
Graph
Use the graph of y = f(x) to solve this exercise.
What are the zeros of f?
-2, 2
Find the value(s) of x for which f(x) = -1.
-1, 1
Find the value(s) of x for which f(x) = -2.
Recall that
the zeros of a function, f, are the x-values for which f(x) = 0.
Example

16. Increasing, Decreasing, and Constant Functions
A function is increasing on an open interval if the graph rises from left to right.
A function is decreasing on an open interval if the graph falls from left to right.
A function is constant on an open interval if it is horizontal (parallel to the x-axis).

17. Example Use the graph of y = f(x) to solve this exercise.
Intervals
Graph
Use the graph of y = f(x) to solve this exercise.
On which interval or intervals is f
increasing?
(-1,2)
decreasing?
(-5, -1) (2, 6)
Example

18. Relative Maxima and Relative Minima
A relative maximum occurs if the graph changes direction from rising to falling on a given interval.
A relative minimum occurs if the graph changes direction from falling to rising on a given interval.

19. Example Use the graph of y = f(x) to solve this exercise.
Relative Maximum
Graph
Use the graph of y = f(x) to solve this exercise.
For what number does f have a relative maximum?
2, f(2) = 5
What is the relative maximum?
(2, 5)
Example

20. Example Use the graph of y = f(x) to solve this exercise.
Relative Minimum
Graph
Use the graph of y = f(x) to solve this exercise.
For what number does f have a relative minimum?
-1, f(-1) = -4
What is the relative minimum?
(-1, -4)
Example

21. Even and Odd Functions
The function f is even if f(-x) = f(x) for all x in the domain of f.
The function f is odd if f(-x) = -f(x) for all x in the domain of f.

22. Example Let Find f(-x) Find –f(x) Neither since f(x) = x2 – x – 4
Even, odd, or neither
Solution
Let
f(x) = x2 – x – 4
Find f(-x)
f(-x) = (-x)2 –(-x) -4
f(-x) = x2 + x – 4
Find –f(x)
-f(x) = -(x2 – x – 4)
-f(x) = -x2 + x + 4
Neither since
f(x) ≠ f(-x) and
f(x) ≠ -f(x)
Example

23. Symmetry
The graph of an even function is symmetric with respect to the y-axis.
The graph of an odd function is symmetric with respect to the origin.

24. Example Use the graph of y = f(x) to solve this exercise.
Even, odd, or neither
Graph
Use the graph of y = f(x) to solve this exercise.
Is f even, odd, or neither?
even
Remember that the function f is an
even function if f(-x) = f(x). The graph is symmetric with respect to the y-axis.
odd function if f(-x) = -f(x). The graph is symmetric with respect to the origin.
Example

25. Graphs of Common Functions
Knowing the graphs of common functions can help you analyze transformations into more complicated graphs.

26. Constant function

27. Identity function

28. Absolute value function

29. Standard quadratic function

30. Square root function

31. Standard cubic function

32. Cube root function

33. Transformations Vertical or horizontal shifts
Reflections about the x- or y-axis
Vertical stretching or shrinking
Horizontal stretching or shrinking

34. Order of Transformations
Horizontal shifting
Stretching or shrinking
Reflecting
Vertical shifting

35. Example Graph g(x) = 4(x – 5)2 + 2
Problem
Solution
Graph g(x) = 4(x – 5)2 + 2
Start with the graph of the basic function f(x) = x2.
Example

36. Example Graph g(x) = 4(x – 5)2 + 2 Look for horizontal shifting.
Problem
Solution
Graph g(x) = 4(x – 5)2 + 2
Look for horizontal shifting.
The x – 5 in parentheses tells us to shift the graph five units to the right.
Example

37. Example Graph g(x) = 4(x – 5)2 + 2
Problem
Solution
Graph g(x) = 4(x – 5)2 + 2
Check for reflecting. This graph doesn’t require reflecting.
Look for stretching or shrinking.
The 4 in front of (x – 5)2 tells us to multiply each y- coordinate by 4, thus vertical stretching the graph.
Example

38. Example Graph g(x) = 4(x – 5)2 + 2
Problem
Solution
Graph g(x) = 4(x – 5)2 + 2
Finally, check for vertical shifting.
The + 2 tells us to raise the graph 2 units.
Observe how the graphs transformed from the red to the gold one.
Example

39. Combinations of Functions
Functions can be
Added
Subtracted
Multiplied
Divided

40. Composition of Functions
Another way of combining two functions.
Denoted by
Defined by

41. Combinations/Composite Functions
Combinations of Functions
Composite Functions
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(fg)(x) = f(x)g(x)
(f/g)(x) = f(x)/g(x)
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))
Combinations/Composite Functions

42. Example Let (x2 – x – 4) – (2x – 6) = x2 - x – 4 – 2x + 6 =
(f – g)(x)
(g – f)(x)
Let
f(x) = x2 – x – 4
g(x) = 2x – 6
(x2 – x – 4) – (2x – 6) =
x2 - x – 4 – 2x + 6 =
x2 - 3x + 2
Let
f(x) = x2 – x – 4
g(x) = 2x – 6
(2x – 6) – (x2 – x – 4)
2x – 6 - x2 + x +4 =
-x2 + 3x - 2
Example

43. Example Let Let f(x) = x2 – x – 4 g(x) = 2x – 6 f(x) = x2 – x – 4
(f ◦ g)(x)
(g ◦ f)(x)
Let
f(x) = x2 – x – 4
g(x) = 2x – 6
Let
f(x) = x2 – x – 4
g(x) = 2x – 6
Example

44. Example Let Let f(x) = 9x - 5 g(x) = (x + 9)/5 f(x) = 9x - 5
(f ◦ g)(x) or f(g(x))
(g ◦ f)(x) or g(f(x))
Let
f(x) = 9x - 5
g(x) = (x + 9)/5
Let
f(x) = 9x - 5
g(x) = (x + 9)/5
Example

45. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
What values are represented by the x- and y-axes?
x is age in months.
y is median height.
Using Modeling

46. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
According to the model, what is the median height of children who are two years old?
Evaluate the function for x = 24.
3.9√ = 25.4 inches
Using Modeling

47. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
How well does the model describe the actual height?
The graph is very close.
Using Modeling

48. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
Use the model to find the average rate of change in inches per month between birth and 12 months.
Use the average rate of change of a function formula
Using Modeling

49. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
Use the model to find the average rate of change in inches per month between 30 and 42 months.
Use the average rate of change of a function formula
Using Modeling

50. Problem
Solution
The function g(x) = 3.9√x models the median height, g(x), in inches, of children who are x months of age.
Compare the two average rates of change.
How is this difference shown on the graph?
The average rate of change is smaller.
The graph is not as steep.
Using Modeling

51. Inverse Functions
If f(g(x)) = x and g(f(x)) = x, function g is the inverse of f.
A one-to-one function has no two different ordered pairs with the same second component. Only one-to-one functions have inverse functions.

52. Horizontal Line Test
A function f has an inverse if there is no horizontal line that intersects the graph of the function f at more than one point.

53. Example Use the graph of y = f(x) to solve this exercise.
Inverse Functions
Graph
Use the graph of y = f(x) to solve this exercise.
Does f have an inverse function? No
Use the Horizontal Line Test to determine if there is an inverse.
Example

54. Finding the Inverse of a Function
Replace f(x) with y in the equation for f(x).
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function.
If f has an inverse function, replace y in Step 3 by f-1(x).
Verify result by showing that f(f-1(x)) = x and f-1(f(x)) = x

55. Example Find the inverse of f(x) = 7x – 5 y = 7x – 5 x = 7y – 5
Solution
Find the inverse of f(x) = 7x – 5
y = 7x – 5
x = 7y – 5
7y = x + 5, y = (x + 5)/7
f-1(x) = (x + 5)/7
Example