precalculus - functions
DESCRIPTION
This is an introductory presentation for precalculus. The topic of the presentation is functions.TRANSCRIPT
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