RELATION … any set of points

RELATION … any set of points

{ (3,6) , (-4,5) , (0,2) }

RELATION … any set of points

{ (3,6) , (-4,5) , (0,2) }

RELATION … any set of points

{ (3,6) , (-4,5) , (0,2) }

RELATION … any set of points

{ (3,6) , (-4,5) , (0,2) }

Remember … A function is a relationship between variables where each input has exactly one output.

Alternate definition = Set of points where each “x” is paired with only one “y”.

Is each of these a function?

{ (2,5) , (7,-1) , (5,4) , (3,8) , (-1,4) , (0,0) }

{ (7, 3) , (5,-2) , (7, 8) , (5,9) , (0, 4) , (6,-3) }

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

( (0,1) , (2,1) , (-2,1) , (-5,1) , (7,1) }

Is each of these a function?

{ (2,5) , (7,-1) , (5,4) , (3,8) , (-1,4) , (0,0) } Yes

{ (7, 3) , (5,-2) , (7, 8) , (5,9) , (0, 4) , (6,-3) } No

{ (1,1) , (1,2) , (1,3) , (1,4) , (1,5) , (1,6) } No

( (0,1) , (2,1) , (-2,1) , (-5,1) , (7,1) } Yes

Is each of these a function?

Is each of these a function?

No No

Yes No

Is each of these a function?

Is each of these a function?

Yes No

Yes No

Is each of these a function?

You can tell whether a graph is a function using the vertical line test.

If a vertical line ever touches the graph more than once, it is NOT a function.

If every vertical line just touches the graph once, it IS a function.

Is each of these a function?

Is each of these a function?

No Yes No No Yes

Domain Set of independent variables in a relation All the x’s that are used

What is the domain?

{ (1,2) , (3,4) , (5,6) , (7,8) , (9,10 }

{ (3,4) , (5,-7) , (-1,2) , (3, 4) , (-1,-1) }

What is the domain?

{ (1,2) , (3,4) , (5,6) , (7,8) , (9,10 } { 1, 3, 5, 7, 9 }

{ (3,4) , (5,-7) , (-1,2) , (3, 4) , (-1,-1) } { 3, 5, -1 }

{1, 2, 3, 4, 5, 6 }

{ 3, 7 }

Range Set of dependent variables in a relation All the y’s that are used

What is the range?

{ (1,2) , (3,4) , (5,6) , (7,8) , (9,10 }

{ (3,4) , (5,-7) , (-1,2) , (3, 4) , (-1,-1) }

What is the range?

{ (1,2) , (3,4) , (5,6) , (7,8) , (9,10 } { 2, 4, 6, 8, 10 }

{ (3,4) , (5,-7) , (-1,2) , (3, 4) , (-1,-1) } { 4, -7, 2, -1 }

{ -1, 0, 1, 2, 3, 6 }

{ 1, 2, 5 }

Function NotationRules or formulas for evaluating functions are often given using notation like …

f(x) = 3x + 4

g(x) = x2

We read f(x) as “f of x”.

f(x) = 3x + 4 means a function of x is defined with the formula 3x + 4

f(x) basically means “y”.

Almost everything in advanced mathematics is defined using function notation. A function basically gives

you a formula that you can

plug numbers into to get an answer.

Typical problem …f(x) = 3x + 4

Find f(7).

Typical problem …f(x) = 3x + 4

Find f(7).Just plug in 7 for x.f(7) = 3 7 + 4

= 21 + 4 = 25

f(x) = 5x – 1 g(x) = 2x

Find f(3) g(4) f(-2) g(-8) f(0)

f(x) = 5x – 1 g(x) = 2x

Find f(3) 14 g(4) 8 f(-2) -11 g(-8) -16 f(0) -1

k(x) = x2 + 3x – 5

Find k(4)

k(x) = x2 + 3x – 5

Find k(4)42 + 34 – 516 + 12 – 5 = 23

p(t) = 180 – 16t2

Find p(3)

p(t) = 180 – 16t2

Find p(3)180 – 16 9 = 36

m(x) = 4x + 3Domain = { 1, 2, 3, 4, 5 }Find the range.

m(x) = 4x + 3Domain = { 1, 2, 3, 4, 5 }Find the range.

Range = { 7, 11, 15, 19, 23 }

g(x) = { (2,5) , (-1,4) , (6,0) }

Find g(2) g(6) g(-1)

g(x) = { (2,5) , (-1,4) , (6,0) }

Find g(2) = 5 g(6) = 0 g(-1) = 4

f(x) =

Find f(-2), f(-1), f(0), f(2)

f(x) =

Find f(-2), f(-1), f(0), f(2)

f(x) = 5x – 1 g(x) = 2x

Find f[g(3)]

g[f(9)]

f(x) = 5x – 1 g(x) = 2x

Find f[g(3)] = f[6] = 29

g[f(9)] = g[44] = 88

g(x) = 5x – 3

If g(x) = 62, what is x?

If g(x) = -98, what is x?

g(x) = 5x – 3

If g(x) = 62, what is x?5x – 3 = 62 x = 13

If g(x) = -98, what is x?5x – 3 = -98

x = -19