Pre-Calculus

Functions and Graphs

What is a function?

Functions in the Real World

What is a function? • A function f is a rule that

assigns to each element x in a Set A exactly one element, called f(x), in a Set B.

• Consider the formula for the

area of a circle in terms of its radius: A = πr2

• For any given value of r, there is only one corresponding value of A. Thus, for example, if the radius of the circle is 3, its area will be 9π.

• Furthermore, the area will always be 9π when the radius is 3. It will not sometimes be 9π and sometimes something else.

• Now consider the formula, B2 = G.

• For any given value of G (except 0), there will be two values of B. Thus, for example, if G = 4, then either B is 2 or B is 2.

• A is a function of r, but B is not a function of G.

Types of functions

• Linear Functions: f(x) = mx + b

• Power Functions: f(x) = xn

• Root Functions: f(x) = (x)1/n

• Reciprocal Functions: f(x) = 1/xn

• Absolute Value Functions: f(x) = lxl

• Greatest Integer Functions: f(x) = ǁxǁ

Types of Functions One-To-One Function

A function from set A to set B is said to be an One-To-One (injective) function if no two or more elements of set A have the same elements mapped or imaged in set B.

Types of Functions

A function from set A to set B is said to be a many-to-one (surjective) function if two or more elements in set A processed through the function produces the same output or same element in set B.

Many-to-one Function

Types of Functions (Summary)

Identifying a Function

• Determine if each of the following are functions.

(a) f(x) = x2 + 1 (b) (f(x))2 = x + 1

Solution

• (a) The first one is a function. Given any value for x which is placed into the function, f(x) gives only one answer.

• (b) The second one is not a function. With certain values for x which are placed into the function, f(x) gives more than one answer.

Identifying a Function

Power Functions

• A function of the form f(x) = xn, where n is any real number constant is called a power function.

• E.g. f(x) = x2 f(x) = x5

Polynomial Functions

• f(x) = anxn + an-1xn-1 + ….. + a1x1 + a0 where n

Є Z+ and a0, a1, …. , an are constants with an ≠ 0 is called a polynomial function in x. The term in which x has the greatest exponent is anxn. This exponent is called degree of the function and an is the leading co-efficient.

• Example f(x) = 3x4 - x3 + 2x2 +5x - 2

Polynomial Functions

Graphs of Polynomial Functions

Identifying a Function Con’t

Identifying a Function Con’t

Rational Functions

• A rational function is a quotient of two polynomials P(x) and Q(x).

• f(x) = P(x) = anxn + an-1xn-1 + … + a2x2 + a1x1+ a0

Q(x) Cnxn + Cn-1xn-1 + … + C2x2 + C1x1 + C0

E.g. f(x) = x2 + 4x + 1 x2 + 3

Evaluating a Function

• Let f(x) = 3x2 + x – 5. Evaluate each function value.

(a) f(4) (b) f(-2) (c) f(½)

Evaluating a Function

• Solution (a) f(4) = 3(4)2 + 4 – 5 f(4) = 3(16) + 4 – 5 f(4) = 47

(b) f(-2) = 3(-2)2 + (-2) – 5 f(-2) = 3(4) + (-2) – 5 f(-2) = 5

Evaluating a Function

• Solution (c) f(½) = 3(½)2 + ½ – 5

f(½) = 3(¼) + ½ – 5 f(½) = -

415

Absolute Value Function

• For any real number x the absolute value or modulus of x is denoted by | x |and is defined as

if x ≥ 0 if x < 0

• The absolute value of a number is always positive or zero.

x

xx

Properties of Absolute Value

Property Example

0a

aa

baab

ba

ba

033

55

5252

312

312

Piecewise Defined Function

• A piecewise defined function is defined by different formulas on different parts of its domain.

• A function f is defined by if x ≤ 1 if x > 1 • Evaluate f(0), f(1) and f(2)

2

1)(

x

xxf

Solution

• Since 0 ≤ 1, we have f(0) = 1 – 0 = 1 • Since 1 ≤ 1, we have f(1) = 1 – 1 = 0• Since 2 > 1, we have f(2) = 22 = 4

Exercises

• Evaluate the function at the indicated values.1. f(x) = 2x + 1;f(1), f(-2), f(½), f(a), f(-a) f(a + b)

2. f(x) = x2 + 2 f(0), f(3), f(-3), f(a), f(-x), f(a-1)

Exercises

3.

f(2), f(-2), f(½), f(a), f(a – 1)

4. f(x) = 2x2 +3x – 4 ;

f(0), f(2), f(-2) f( ), f(x +1), f(-x)

;11)(

xxxf

2

Exercises

5. f(x) = 2|x – 1|;

f(-2), f(0), f(½), f(2), f(x + 1), f(x2 + 2)

6. f(x) = x3 – 4x2;

f(0), f(1), f(-1), f(⅔), f(x2)

Exercises

• Evaluate the piecewise defined function at the indicated values.

if x ≤ -1 1. if -1 < x ≤ 1 if x > 1

f(-4), f( ), f(-1), f(0), f(25)

1

2)(

2

xxx

xf

23

Exercises

if x < -1 2. if 0 ≤ x ≤ 2 if x > 2

f(-5), f(0), f(1), f(2), f(5)

2)2(

13

)(

x

xx

xf

The Domain of a Function

• The domain of a function is the set of all 'allowable’ inputs for the function.

• The domain of the function may be stated explicitly.

E.g. f(x) = x2 0 ≤ x ≤ 5

The Domain of a Function

• On the other hand, the domain of a function may not be stated explicitly.

E.g. f(x) = 5x – 3

• Find the set of all values that can be plugged into a function and have the function exist and have a real number for a value.

The Domain of a Function • Therefore, for the domain we need to avoid:

division by zero, square roots of negative numbers, logarithms of zero and logarithms of negative numbers, etc.

• Find the domain of the function; f(x) = 5x – 3

• Domain: or , x

The Range of a function

• The range of a function is the set of all outputs for the function.

E.g. Let y = f(x) = |2x - 1|

• Range: or [ 0, ∞)

Expressing the Domain & Range of a Function

• Consider the function y = f(x) = x2

Domain Range

{ x : - ∞ < x < ∞ } { y : y ≥ 0 }

, ,0

00

Set Notation

• If a and b are fixed real numbers with a < b we define the following sets;

1. is an infinite unbounded interval. This is described as an open left interval.

2. is an infinite unbounded interval which

contains the end point.

axx :

axx :

Set Notation

3. is an open interval.

4. is a closed interval.

5. is a half-open interval.

6. is a half-open-interval.

bxax :

bxax :

bxax :

bxax :

Set Notation & Symbol Representation

,: aaxx aaxx ,:

,: aaxx aaxx ,:

babxax ,: babxax ,:

babxax ,: babxax ,:

Finding Domains of Functions

• Examples

1.

Solution: the function f is not defined at x = 4, therefore its domain is { x : x ≠ 4 }.

Domain: ( - ∞, 4) (4, ∞)

41)(

x

xf

Finding Domains of Functions

• Examples

2.

Solution: the function f is not defined for negative x, therefore its domain is { x : x ≥ 0 }.

Domain: [0 , ∞ )

xxf )(

Finding Domains of Functions

• Example

3.

Solution: the function f is undefined if x = -3, x = 5

Domain: or

1524)( 2

xxxxf

01522 xx

5,3: xxx ,53,

Finding Domains of Functions

• Example

4.

Solution: the function f is undefined if 6 + x - x2 ≥ 0

Domain: { x : - 2 ≤ x ≤ 3 } or [-2,3]

26)( xxxf

Finding Domains & Ranges of Functions

• Examples

1. Let y = f ( z) = |z - 6| - 3

Solution: the function f is an absolute function and is defined for any real value.

Finding Domains & Ranges of Functions

• Domain: { z : - ∞≤ z ≤ ∞ } or (- ∞,∞)

• Range: or

Finding Domains & Ranges of Functions

• Examples

2. Let y =

Solution : the function f is not defined for square root of any negative real values.

ttf 74)(

Finding Domains & Ranges of Functions

• Domain: or

• Range: or

Finding Domains and Ranges of Functions

• Examples

3.

Solution: the function f is a constant function and is defined at 8.

Finding Domains and Ranges of Functions

• Domain: or

• Range: 8

Exercises

• Find the domain of the following functions.

Exercises