FunctionsSection 1.4

Relation

• The value of one variable is related to the value of a second variable

• A correspondence between two sets• If x and y are two elements in these sets and if

a relation exists between x and y, then x corresponds to y, or y depends on x, written x y

• x is the input to the relation and y is the output of the relation

Ways to Express Relations

• Equations• Graphs• Mapping– Use a set of inputs and draw arrows to the

corresponding element in the set of outputs– See page 31

Function

• A relation that associates with each element of a domain exactly one element in the range (called the value or image)

• Denoted by letters such as f, F, g, G, etc.• To determine if a relation is a function, solve

equation for y– More than one solution for y: not a function– Otherwise it’s a function

Page 43

15.

19.

33.

Find the Value of a Function

• f(x)– f of x– f at x– The value of f at the number x

• x is the independent variable (argument)• y is the dependent variable

For the function f defined by f(x) = x2 + 4x, evaluate

(A) f(3)

(B) f(x) + f(3)

(C) f(-x)

For the function f defined by f(x) = x2 + 4x, evaluate

(D) -f(x)

(E) f(x + 3)

For the function f defined by f(x) = x2 + 4x, evaluate

(F)

€

f (x + h) − f (x)

h, h ≠ 0

is called the difference quotient of f€

f (x + h) − f (x)

h, h ≠ 0

When a function is defined by an equation in x and y, the function f

is given implicitly.

If it’s possible to solve the equation for y in terms of x, write y = f(x),

which means the function is given explicitly.

Pages 43-44 (24-46 even)

Find the Domain of a Function

• The domain of f is the largest set of real numbers for which the value f(x) is a real number.

• If x is in the domain of a function, f is defined at x, or f(x) exists.

• If x isn’t in the domain of a function, f is not defined at x, or f(x) does not exist.

€

f (x) = x 3 + 4

€

g(x) =4x

x 2 −16

€

h(t) = 5 −10t

€

V (s) = s3

Sum of two functions

€

( f + g)(x) = f (x) + g(x)

Difference of two functions

€

( f − g)(x) = f (x) − g(x)

Product of two functions

€

( f ⋅g)(x) = f (x) ⋅g(x)

Quotient of two functions

€

f

g

⎛

⎝ ⎜

⎞

⎠ ⎟(x) =

f (x)

g(x) g(x) ≠ 0

Page 44

61.

page 44(48, 52, 56, 58, 62-70 even, 73-78)

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pages 44-45(50, 54, 60, 80, 82, 84, 86, 87, 89)