Lesson 1.2, pg. 138Functions & Graphs

Objectives: To identify relations and functions, evaluate functions, find the domain and range of

functions, determine whether a graph is a function, and graph a function.

Domain & Range

• A relation is a set of ordered pairs.• Domain: first components in the relation

(independent); x-values• Range: second components in the relation

(dependent, the value depends on what the domain value is); y-values

• Find the domain and range of the relation. {(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}

FUNCTIONS

• Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.

Every “x” has only one “y”.

Mapping – illustrates how each member of the domain is paired with each member of the range (Note: List

domain and range values once each, in order.)

x y

0457

-912

Is this relation a function?

Draw a mapping for the following. (5, 1), (7, 2), (4, -9), (0, 2)

See Example 2, page 150.

Determine whether each relation is a function:A) {(1,2), (3,4), (5,6), (5,8)}

B) {(1,2), (3,4), (6,5), (8,5)}

Functions as EquationsDetermine whether the equation defines y as a function of x.

a) b)

1.Solve for y in terms of x.2.If two or more values of y can be obtained for a given x, the equation is not a function.

44 222 yxyx

Determine if the equation defines y as a function of x.

A) 2x + y = 6

B) x2 + y2 = 1

C) x2 + 2y = 10

Evaluating a Function

• Common notation: f(x) = function

• Evaluate the function at various values of x, represented as: f(a), f(b), etc.

• Example: f(x) = 3x – 7 f(2) = f(3 – x) =

If f(x) = x2 – 2x + 7, evaluate each of the following.

• a) f(-5) b) f(x + 4) c) f(-x)

See Example 4, page 143 for additional practice.

Determine if a relation is a function from the graph?

• Remember: to be a function, an x-value is assigned ONLY one y-value .

• On a graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line.

• THUS, the vertical line test! If a vertical line drawn on any part of your graph touches more than one point, it is NOT the graph of a function.

Graphs of Functions

Step 1: Graph the relation. (Use graphing calculator or pencil and paper.)

Step 2: Use the vertical line test to see if the relation is a function.

• Vertical line test – If any vertical line passes through more than one point of the graph, the relation is not a function.

Determine if the graph is a function.

a) b) y

x

5

5

-5-5

x

y

Here’s more practice.

c) d) y

x

y

x

Example

Analyze the graph.2( ) 3 4

a. Is this a function?

b. Find f(4)

c. Find f(1)

d. For what value of x is f(x)=-4

f x x x

x

y

(a)

(b)

(c)

(d)

Find f(7).

x

y

0

1

1

2

Can you identify domain & range from the graph?

• Look horizontally. What x-values are contained in the graph? That’s your domain!

• Look vertically. What y-values are contained in the graph? That’s your range!

• Write domain and range using interval or set-builder notation.

• See Example 8, page 148.

Domain: set of all values of xRange: set of all values of y

•Always write the domain and range in interval notation when reading the domain and range from a graph.•Use brackets [ or ] to show values that are included in the graph.•Use parentheses ( or ) to show values that are NOT included in the graph.

x

yIdentify the function's domain and range from the graph

Domain (-1,4]

Range [1,3)

Domain [3, )

Range [0, )

x

y

Example

Identify the Domain and Range from the graph.

x

y

Example

Identify the Domain and Range from the graph.

x

y

Example

Identify the Domain and Range from the graph.

x

y

(a)

(b)

(c)

(d)

Find the Domain and Range.

D:(- , ) R:(-5,7]

D:(-5, ) R: (- , )

D:(- , ) R: [-5, )

D:[- , ] R: [-5, ]

x

y

What is the difference in the two sets below, and when should we use each to describe the domain of a function?

{1,2,3,4} [1,4]

Finding intercepts:

• x-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero.

• y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero.

• Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!

We can identify x and y intercepts from a function's graph.

To find the x-intercepts, look for the points at which the graph

crosses the x axis. The y-intercepts are the points where the graph

crosses the y axis.

The zeros of a function, f, are the x values for which f(x)=0.

These are the x intercepts.

By definition of a function, for each value of x we can

have at most one value for y. What does this mean in terms

of intercepts? A function can have more than one x-intercept

but at most one y intercept.

Example

Find the x intercept(s). Find f(-4)

x

y

x

y

Example

Find the x and y intercepts. Find f(5).

Summary

• Domain = x values• Range = y values• Use the vertical line test to verify if a graph is

a function.• To evaluate means to substitute and simplify.• Intercepts – where function crosses the x-or y-

axis