Pre-Calculus Mathematics 12 - 1.1 - Functions and ...

Pre-Calculus Mathematics 12 - 1.1 - Functions and Relations

Goals: 1. Identify relations, functions, one-to-one functions, domains, ranges, vertical and horizontal line tests, restrictions 2. Recognize function types Definitions: Domain - the set of all possible ( ) values of a relation. Range - the set of all possible ( ) values of a relation. Relation - a set of ordered pair(s) Function - a relation in which each domain ( ) value is paired with only one unique range ( ) value. Vertical line test - an equation defines y as a function of x if and only if every vertical line in the coordinate plane intersects the graph of the equation only once.

Example 1: Determine the domain/range of the following graphs and whether they are a function/relation

Types of Functions: 1. Constant function: - eg. x = k

2. Linear: y mx b or f x mx b

3. Quadratic Standard form

Pre-Calculus Mathematics 12 - 1.1 - Functions and Relations General (expanded) form

f (x) a(x h)2 k

f (x) ax2 bx c a,b, c a 0

vertex Axis of symmetry: if a > 0, graph opens ______ if a < 0, graph opens ______

4. Cubic: f (x) ax3 + bx2 cx d

5. Absolute value: f (x) a x h k

6. Radical: f (x) a x h k

7. Reciprocal:

f (x) a k xh

Pre-Calculus Mathematics 12 - 1.1 - Functions and Relations

Restrictions on the domain of a functions: 1. Cannot have a negative number inside an even root.

f (x) 3 x

2. Cannot have zero in a denominator

f (x) 8 x6

One?to?One Function A one-to-one function is a function in which every single value of the domain is associated with only one value in the range, and vice-versa.

Horizontal line test - for a one-to-one function: A function, f(x), is a one-to-one function of x if and only if every horizontal line in the coordinate plane intersects the function only once at most.

Example 2: Determine whether the following relations are functions, one-to-one functions or neither

2

A

4

B

6

8

C

10

D

1

A

2

B

3

C

4

D

5

E

1

A

2

B

3

C

4

D

5

E

A

B

1

C

D

E

Practice: Pg 9 & 10 # 1, 2 and 3

Pre-Calculus Mathematics 12 - 1.2 - Arithmetic Combinations of Functions

Goal: 1. Perform operations with functions both with and without the graph of the function

Functions are number generators. When you put a value for the domain in the function, you will get the resulting value in the range.

f x x2 5

And just like numbers, functions can be added, subtracted, multiplied and divided.

Use the following functions, f x 2x ?1 & g x x2 ? 4 to determine:

a) Sum: f g x f (x) g(x)

b) Difference: f g x f (x) g(x)

Product: fg x f (x) g(x)

Quotient:

f g

x

f (x) g(x)

Note: The domain of the new function must include the restrictions of the new functions as well as the restriction(s) of the original function(s)

Example 1: Given functions below, determine each new combined function and its domain.

1 f (x) x2 ,

g(x) 2 , h(x) 2x2 5x 3 i(x) x2 9, x 2

j(x) x2, k(x) x

a) gj3

b)

f g

4

Pre-Calculus Mathematics 12 - 1.2 - Arithmetic Combinations of Functions Example 1 continued... Given functions below, determine each new combined function and its domain.

f

(x)

1 x2

,

g(x) 2 , h(x) 2x2 5x 3 i(x) x2 9, x2

j(x) x2, k(x) x

c) g f x

d)

gk i

x

e)

1 k

1 k

x

f)

f x

x

Pre-Calculus Mathematics 12 - 1.2 - Arithmetic Combinations of Functions Example 1 continued... Given functions below, determine each new combined function and its domain.

f

(x)

1 x2

,

g(x) 2 , h(x) 2x2 5x 3 i(x) x2 9, x2

j(x) x2, k(x) x

g)

g f

x

h i

x

h) g h i x

Pre-Calculus Mathematics 12 - 1.2 - Arithmetic Combinations of Functions

Example 2: Use the graph of f x and g x to graph the following functions.

f x g x a) f g x

b)

2

f

1 2

g

x

Practice: Pg 13 -16 # 1, 2, 3 (a, b, c, d, i, j), 4 (a, d), 5b, 6c, 7d, 8e

Pre-Calculus Mathematics 12 - 1.3 ? Composite Functions

Goals: 1. Perform the composite of two or more functions 2. Decompose a composite function

A function can be considered like a machine. It has an input value (x) and generates an output, y, or f (x) .

f x 3x ? 4

A function can also be "input" into another function. This would then generate a composite function.

Using the functions f x =3x ? 4 & g x x2 2

Determine: f g x

g f x

Notation: f g x ( f g)(x) f g

With composite functions, the output of one function becomes the input of another function(s). When determining ( f g)(x) the output of function g(x) becomes the input of the function f (x) .

Example 1: Given functions below, determine each composite function.

f

(x)

1 x2

,

g(x) 2 , h(x) 2x2 5x 3, x2

i(x) x2 9, k(x) x

a) (h g)(3)

b) (k g f )(2)

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