algebra 1 glencoe mcgraw-hill malinda young relations, functions & function notation

Algebra 1 Glencoe McGraw-Hill Malinda Young

Relations, Functions& Function Notation

A relation is a set of ordered pairs.

A relation can be represented by a graph, a table, or a mapping.

Ordered Pairs

Table

Graph

Mapping

(4, 3) (-2, -1) (-3, 2) (2, -4) (0, -4)

x y4 3-2 -1-3 22 -40 -4

4-2-3 2 0

3-1 2-4

x y

Express the relation { (-1, 0), (2, -4), (-3, 1), (4, -3)} as a table, a graph, and a mapping. Then

determine the domain and range.

x y-1 0 2 -4-3 1 4 -3

-1 2-3 4

0 -4 1 -3

You should recall that the domain is the set of all inputs (x) and the range is the set of all

outputs (y).Domain: {-1, 2, -3, 4} Range: {0, -4, 1, -3}

The inverse of any relation can be found by switching the inputs and the

outputs of each ordered pair.

The domain of the relation becomes the range of the inverse.

The range of the relation becomes the domain of the inverse.

5740

1 2-9

1 2-9

5740

Relation Inverse of the Relation

(5, 1)(7, 2)(4, -9)(0, 2)

(1, 5)(2, 7)(-9, 4)(2, 0)

(5, 1)(7, 2)(4, -9)(0, 2)

How does the graph of a relation compare with the graph of the inverse of the

relation?Graph the points of this relation, then

connect the points:(4, 3) (-2, -1) (-3, 2)

(2, -4) (0, -4)(3, 4) (-1, -2) (2, -3)

(-4, 2) (-4, 0)

Graph the points of the inverse of the relation,

then connect the points:

Write an observation in your notes.

A relation is any set of ordered pairs (x, y).

A function is a rule that establishes a relationship between two quantities, called

the input and the output.

For each input, there is exactly one output – even though two different inputs may give

the same output.

Huh?

That is what’s meant by “even though different inputs” (medium Coke, medium

Sprite) “may give the same output” (same price for both items).

You notice also that other medium sodas like Sprite also cost the same as a medium Coke.

Let’s say you go to Baja Fresh. With your lunch you order a medium Coke. Will you see two different prices for a medium Coke on the

menu? No.That is what’s meant by “for each input” (medium Coke), “there is

exactly one output” (one price for the medium Coke).

One way to look at functions is to make an input/output chart like the one below.

This relation is a function. For every x-value (input) there is exactly one y-

value (output).

x y1 02 -43 14 -35 106 12

Is this relation a function?

No, it is not a function. Why? Because the input 2 had two

different outputs. Think of it like this:

2 Cokes can’t be $4 and $10.

x y1 22 41 24 82 106 12

Is this relation a function?

Domain: {1, 2, 3, 4, 5, 6}Range: {4, 6, 8, 10, 12, 14}

x y1 42 63 84 105 126 14

Does the table represent a function? Explain. If it is a function, name the domain and the

range.The table DOES

represent a function because for every

input there is only one output.

Domain: {-3, -2, -1} Range: {4, 6, 8}

The table DOES represent a function because for every

input there is only one output.

x y-3 4-2 6-1 8-3 4-2 6-1 8

Does the table represent a function? Explain. If it is a function, name the domain and the

range.

Does the table represent a function? Explain. If it is a function, name the domain and the

range.x y5 37 52 09 75 47 5

No, it is not a function. Why? Because the input 5 had two

different outputs. Think of it like this:

5 Cokes can’t be $3 and $4.

Does the table represent a function? Explain. If it is a function, name the domain and the

range.x y-3 8-2 6-1 4 0 2 1 4 2 6

Domain: {-3, -2, -1, 0, 1, 2} Range: {2, 4, 6, 8}

The table DOES represent a function because for every

input there is only one output.

Function Notation: When a function is defined by an equation, it is often convenient to name the function. Just as x is commonly used as a variable, the letter f is commonly

used to name a function.

To use function notation, replace y with f(x).

The symbol f(x) is read as “the value of f at x” or simply “f of x”. It does not mean f times x.

x-y notation

function notation

2x3y 2x3)x(f

Any letter can be used to name a function. If you see f(x), g(x), h(x) or

any other variable it is the name for the function and it is used in place of y.

In a function, x represents the elements of the domain

andf(x) represents the elements of the

range.

If f(x) = 2x + 3, find the value of the function.

)6(f You are being asked to replace the x in the function with -6.

3)6(2)6(f Replace x with -6.

312 Multiply.

Add.9

In this function, when x = -6, f(x) = -9.

If f(x) = 2x + 3, find the value of the function.

)11(f You are being asked replace the x in the function with 11.

3)11(2)11(f Replace x with 11.

322 Multiply.

Add.25

In this function, when x = 11, f(x) = 25.

If f(x) = x2 – 7x + 3, find the value of the function.

You are being asked to replace the x in the function with 1.

)1(f

3)1(7)1()1(f 2 Replace x with 1.371 Multiply.

Add.3

In this function, when x = 1, f(x) = -3.

If g(x) = x2 + x + 3,find the value of the function.

You are being asked to replace the x in the function with 4n.

)n4(g

3)n4()n4()n4(g 2 Replace x with 4n.3n4n16 2 Multiply.

In this function, when x = 4n, g(x) = 16n2 + 4n + 3

If h(x) = x2 + 1find the value of the function.

You are being asked to replace the x in the function with n.

)n(h

1)n()n(h 2 Replace x with n.

1n2

In this function, when x = n, h(x) = n2 + 1

If f(x) = 5x + 2,find the value of the function.

You are being asked to replace the x in the function with r. The 4 outside the brackets means that the entire function will be multiplied

by 4.

)r(f4

2)r(54)r(f4 Replace x with r.

2r54

In this function, when x = r, 4[f(r)] = 20r + 8

8r20 Multiply all terms by 4.

A vertical line test is used to determine whether a graph represents a function. A graph is a function if any vertical line

intersects the graph at no more than one point.

x (input)

y (output)

The graph is a function.

You can use your pencil to check if a graph is a function. Keep your pencil straight to represent a

vertical line and pass it across the graph. If it touches the graph at more than one point, the graph is not a

function.

x (input)

y (output)

x (input)

y (output)

The graph is NOT a function.

x (input)

y (output)

The graph is NOT a function.

x (input)

y (output)

The graph IS a function.

x (input)

y (output)

The graph is NOT a function.

x (input)

y (output)

The graph is a function.

x (input)

y (output)

The graph is a function.

Determine whether the relation is a function.

y = 2x + 1x y-2-1 0 1 2

.....

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