5.2 Relations and Functions• A relation is a set of ordered pairs.

• The domain of a relation is the set of first coordinates of the ordered pairs – the x-coordinates.

• The range of a relation is the set of second coordinates – the y-coordinates.

Finding Domain and Range• Find the domain and range of the ordered pairs listed

for the giraffe data.

(18 , 4.25)(20 , 4.40)

(21 , 5.25)

(14 , 5.00)

(18 , 4.85)

Domain: {14, 18, 20, 21}

Range: {4.25, 4.40, 4.85, 5.00, 5.25}

Age (years)

Height (meters)

18 4.25

20 4.40

21 5.25

14 5.00

18 4.85

Giraffes Age Height

Function

• A function is a relation that assigns exactly one value in the range to each value in the domain.– You can tell if a relation is a function by

analyzing the graph of a relation using the vertical-line test.

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

Using the Vertical-Line Test• Determine whether the relation {(3 , 0), (-2 , 1),

(0 , -1), (-3 , 2), (3 , 2)} is a function.

• Step 1 – graph the ordered pairs on a coordinate plane.

Vertical-Line Test• Step 2 – use the Vertical – Line test.

• A vertical line passes through both (3 , 0) and (3 , 2), so the relation is not a function.

Using a Mapping Diagram• Determine whether each relation is a function.

a. {(11 , -2) , (12 , -1) , (13 , -2) , (20 , 7)}

Domain Range

11 -2

12 -1

13 7

20

There is no value in the domain that corresponds to more than one value of the range.

The relation is a function.

Using a Mapping Diagram• Determine whether each relation is a function.

b. {(-2 , -1) , (-1 , 0) , (6 , 3) , (-2 , 1)}

Domain Range

-2 -1

-1 0

6 1

3

The domain value corresponds to two range values -1 and 1.

The relation is not a function.

Evaluating Functions• A function rule is an equation that describes a

function.– The domain is the set of input values.– The range is the set of output values.

• A function is in function notation when you use f(x) to indicate the outputs.– You read f(x) as “f of x” or “f is a function of x”.– The notations g(x) and h(x) also indicate functions

of x.

Evaluating a Function Rulea. Evaluate f(n) = -3n – 10 for n = 6.

f (n) = -3n – 10

f (6) = -3(6) – 10

f (6) = -18 – 10

f (6) = -28

b. Evaluate y = -2x2 + 7 for x = -4

y = -2(-4)2 + 7

y = -2(16) + 7

y = -32 + 7

y = -25

Finding the Range• Evaluate the function rule f(a) = -3a + 5 to find

the range of the function for the domain {-3 , 1 , 4}.

a. f(a) = -3a + 5 f(-3) = -3(-3) + 5 f(-3) = 14

b. f(a) = -3a + 5 f(1) = -3(1) + 5

f(1) = 2c. f(a) = -3a + 5

f(4) = -3(4) + 5 f(4) = -7

More Practice!!!

• Textbook – p. 244 #2 – 26 even.