08 functions and their graphs - part 1 - revised

Functions and Their Graphs

Definition of Relation Relation – a set of ordered pairs, which

contains the pairs of abscissa and ordinate. The first number in each ordered pair is the x-value or the abscissa, and the second number in each ordered pair is the y-value, or the ordinate.

  Domain is the set of all the abscissas, and

range is the set of all ordinates.

Relations A relation may also be shown using a table of

values or through the use of a mapping diagram. Illustration: Using a table. Using a mapping

diagram.Domain

Range

0 11 22 33 44 57 8

Definition of Function Function – a characteristic of set of

values where each element of the domain has only one that corresponds with it in the range. It is denoted by any letter of the English alphabet.

The function notation f(x) means the value of function f using the independent number x.

Example 1a. Given the ordered pairs below, determine

if it is a mere relation or a function.

(0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)

Answer: For every given x-value there is a

corresponding unique y-value. Therefore, the relation is a function.

Example 1b. Which relation represents a function?A. {(1,3), (2, 4), (3,5), (5, 1)}B. {(1, 0), (0,1), (1, -1)}C. {(2, 3), (3, 2), (4, 5), (3, 7)}D. {(0, 0), (0, 2)}

Answer: A

Example 1c. Which mapping diagram does not represent a

function? A. B.

C. D.

Evaluation of Functions

Example 2.

If f (x) = x2 + 3x + 5, evaluate:

a. f (2) b. f (x + 3) c. f (-x)

Solution a. We find f (2) by substituting 2 for x in the equation.f (2) = 22 + 3 • 2 + 5 = 4 + 6 +

5 = 15

Thus, f (2) = 15.

Example 2.

Solutionb. We find f (x + 3) by substituting (x + 3) for x in the equation.

f (x + 3) = (x + 3)2 + 3(x + 3) + 5

If f (x) = x2 + 3x + 5, evaluate: b. f (x + 3)

Equivalently,

f (x + 3) = (x + 3)2 + 3(x + 3) + 5 = x2 + 6x + 9 + 3x + 9 + 5 = x2 + 9x + 23.

Example 2.

Solutionc. We find f (-x) by substituting (-x) for x in the equation.

f (-x) = (-x)2 + 3(-x) + 5

If f (x) = x2 + 3x + 5, evaluate: c. f (-x)

Equivalently,

f (-x) = (-x)2 + 3(-x) + 5 = x2 –3x + 5.

Example 3a. Which is the range of the relation

described by y = 3x – 8 if its domain is {-1, 0, 1}?

A) {-11, 8, 5} B) {-5, 0 5} C) {-11, -8, -5} D) {0, 3, 5}

Example 3b. Which is the range of the relation

described by 3y = 2x2 – 36 if its domain is {3, 6, 9}?

A) {-6, 12, 42} B) {6, 12, 42} C) {0, 6, 12} D) {-6, 0, 12}

Operations on Functions

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions. The sum, the difference, the product , and the quotient are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows:

Sum: (f + g)(x) = f (x)+g(x)

Difference: (f – g)(x) = f (x) – g(x)

Product: (f • g)(x) = f (x) • g(x)

Quotient: (f / g)(x) = f (x)/g(x), g(x) ≠ 0

Example 4a.Let f(x) = 2x+1 and g(x) = x2 - 2. Find a. (f + g) (x) c.(g – f) (x) e. (f / g) (x)b. (f – g) (x) d. (f ∙ g) (x) f. (g/f) (x)Solution:a. (f + g) (x) = f(x) + g( x) = (2x+1 )+ (x2 – 2) = x2 +

2x - 1b. (f – g)(x) = f(x) - g(x) = (2x+1) - (x2 - 2) = -x2 + 2x +

3c. (g – f)(x) = g(x) - f(x) = (x2 - 2) – (2x +1) = x2 - 2x - 3d. (f ∙ g)(x) = f(x) ∙ g(x) = (2x+1)(x2 - 2) = 2x3 + x2 - 4x

- 2e. (f/g)(x) = f(x)/g(x) = (2x+1)/(x2 - 2), f. (g/f)(x) = g(x)/f(x) = (x2 - 2)/(2x +1),

2x

2

1x

Example 4b.Let f(x) = 3x+6 and g(x) = x +2. Find a. (f + g) (1)b. (f – g) (2)c. (f ∙ g) (0)d. (f/g) (-1)e. (g/f) (-1)ANSWERS:f. (f + g) (1) = 12g. (f – g) (2) = 8h. (f ∙ g) (0) = 12i. (f/g) (-1) = 3j. (g/f) (-1) = 1/3

The Composition of Functions

The composition of the function f with g is denoted by f o g and is defined by the equation

(f o g)(x) = f (g(x)).

The domain of the composite function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f.

Example 5a.Given f (x) = 3x – 4 and g(x) = x2 + 6,

find: a. (f ○ g)(x) b. (g ○ f)(x) Solution a. We begin with (f o g)(x), the composition of f with g.

Because (f o g)(x) means f (g(x)), we must replace each occurrence of x in the equation for f by g(x).

f (x) = 3x – 4(f ○ g)(x) = f (g(x)) = 3(g(x)) – 4

= 3(x2 + 6) – 4 = 3x2 + 18 – 4 = 3x2 + 14

Thus, (f ○ g)(x) = 3x2 + 14.

Solution b. Next, we find (g o f )(x), the composition of g with

f. Because (g o f )(x) means g(f (x)), we must replace each occurrence of x in the equation for g by f (x). g(x) = x2 + 6

(g ○ f )(x) = g(f (x)) = (f (x))2 + 6 = (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6 = 9x2 – 24x + 22

Notice that (f ○ g)(x) is not the same as (g ○ f )(x).

Example 5a.Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f ○ g)(x) b. (g ○ f)(x)

Example 5b.Given f (x) = x – 2 and g(x) = x + 7, find: a. (f ○ g)(x)

b. (g ○ f)(x)

c. (f ○ f)(x)

d. (g ○ g)(x)

Answers:a. (f ○ g)(x) = x + 5b. (g ○ f)(x) = x + 5

c. (f ○ f)(x) = x - 4

d. (g ○ g)(x) = x + 14

Graphs of Relations and

Functions

Graph of a Function If f is a function, then the graph of f is the set of

all points (x,y) in the Cartesian plane for which (x,y) is an ordered pair in f.

The graph of a function can be intersected by a vertical line in at most one point.

Vertical Line Testo If a vertical line intersects a graph more than once, then

the graph is not the graph of a function.

Example 6a.

Determine if the graph is a graph of a function or just a graph of a relation.

8

6

4

2

-2

-4

5 10 15

graph of a relation

Example 6b.

Determine if the graph is a graph of a function or just a graph of a relation.

graph of a function

Example 6c.

Determine if the graph is a graph of a function or just a graph of a relation.

graph of a relation

Example 6d.

16

14

12

10

8

6

4

2

2

4

6

8

15 10 5 5 10 15 20 25

A

Determine if the graph is a graph of a function or just a graph of a relation.

graph of a relation

Example 6e.

Determine if the graph is a graph of a function or just a graph of a relation.

4

3

2

1

-1

-2

-3

-4

-6 -4 -2 2 4 6

graph of a relation

Example 6f.

6

4

2

-2

-4

-6

-10 -5 5 10

Determine if the graph is a graph of a function or just a graph of a relation.

graph of a relation

Example 6g.

Determine if the graph is a graph of a function or just a graph of a relation.

graph of a function

3 1 -3 -2 -1 1 2 3 4 -1 -2 -3 -5

Finding Domain and Range

from Graphs

Example 7a.

Domain: • The set of real

numbers• (-∞, ∞)• x

Range: • The set of real

numbers• (-∞, ∞)• y

Example 7b.

Domain: • The set of real

numbers• (-∞, ∞)• x

Range: • [-1, ∞)• y -1

Example 7c.

16

14

12

10

8

6

4

2

-20 -15 -10 -5 5 10

Domain: • The set of real

numbers• (-∞, ∞)• x

Range: • [0, ∞)• y 0

Example 7d.

Domain: • (-∞, 0) (0, ∞)• The set of real

numbers, except 0

Range: • (-∞, -1) (1, ∞)• The set of real

numbers except [-1,1]

Types of Functions

1.Increasing, Decreasing, and Constant Functions

Constantf (x1) < f

(x2)

(x1, f (x1))

(x2, f (x2))

Increasingf (x1) < f

(x2)

(x1, f (x1))

(x2, f (x2))

Decreasingf (x1) < f

(x2)

(x1, f (x1))

(x2, f (x2))

A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2).A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2). A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2).

Solutiona. The function is decreasing on the interval (-∞, 0),

increasing on the interval (0, 2), and decreasing on the interval (2, ∞).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

-5

-4

-3

-2

-1

1 2 3 4 5

5

4

3

1

-1-2-3-4-5

-5

-4

-3

-2

-1

1 2 3 4 5

5

4

3

2

1

-1-2-3-4-5

a.

b.

Example 8a.

Solution: b.• Although the function's equations are not given, the graph

indicates that the function is defined in two pieces. • The part of the graph to the left of the y-axis shows that

the function is constant on the interval (-∞, 0). • The part to the right of the y-axis shows that the function

is increasing on the interval [0,∞).

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown.

-5

-4

-3

-2

-1

1 2 3 4 5

5

4

3

1

-1-2-3-4-5

-5

-4

-3

-2

-1

1 2 3 4 5

5

4

3

2

1

-1-2-3-4-5

a.

b.

Example 8a.

Example 8b.

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Decreasing on (-∞, 0);Increasing on (0, ∞)

Example 8b.

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Increasing on (-∞, 2);Constant on (2, ∞)

Example 8c.

Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Increasing on (-∞,∞)

2.Continuous and Discontinuous Functions

A continuous function is represented by a graph which may be drawn using a continuous line or curve, while a discontinuous function is represented by a graph which has some gaps, holes or breaks (discontinuities).

3.Periodic Functions

A periodic function is a function whose values repeat in periods or regular intervals.

y = tan(x) y = cos(x)

A linear function is a function of the form f(x) = mx +b where m and b are real numbers and m ≠ 0.Domain: the set of real numbersRange: the set of real numbersGraph: straight lineExample: f(x) = 2 - x

4. Linear Functions

5. Quadratic Functions

A quadratic function is a function of the form f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0.Domain: the set of real numbersGraph: parabolaExamples: parabolas parabolas

opening upward opening downward

The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph below. If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward. The vertex (or turning point) is the minimum or maximum point.

Graphs of Quadratic Functions

Graphing Parabolas

Given f(x) = ax2 + bx +c1. Determine whether the parabola opens upward or

downward. If a > 0, it opens upward. If a < 0, it opens downward.

2. Determine the vertex of the parabola. The vertex is

3. The axis of symmetry is

The axis of symmetry divides the parabola into two equal parts such that one part is a mirror image of the other.

a

bac

a

b

4

4,

2

2

a

bx

2

Graphing Parabolas

Given f(x) = ax2 + bx +c4. Find any x-intercepts by replacing f (x) with 0.

Solve the resulting quadratic equation for x. 5. Find the y-intercept by replacing x with zero. 6. Plot the intercepts and vertex. Connect these

points with a smooth curve that is shaped like a cup.

Example 9. The function f(x) = 1 - 4x - x2 has its

vertex at _____.  A. (2,11) B. (2,-11) C.( -2,-3) D.(-2,1)

Example 10a. Identify the graph of the given function: y = 3x2 -

3.

Example 10b. Identify the graph of the given function: 4y = x2.

Example 10c. Identify the graph of the given function: y = (x -

2)(x – 2).

Minimum and Maximum: Quadratic Functions

Consider f(x) = ax2 + bx +c.

1. If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)).

2. If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)).

Example 11a.

The maximum value of the function f(x) = -3x2 – 2x + 4 is ____.

  A. 13/3 B. 3/13 C. 9 D. 13

Example 10b.

The function f(x) = x2 – 8x + 16 has _____.  A. a minimum value at x = -4 B. a maximum value at x = -4 C. a minimum value at x = 4 D. a maximum value at x = 4

References: http://rechneronline.de/function-graphs/ http://www.coolmath.com/graphit/