B. FunctionsCalculus 30

1. Introduction• A relation is simply a set of ordered pairs.

• A function is a set of ordered pairs in which each x-value is paired with one and only one y-value.

• Graphically, we say that the vertical line test works.

• can be written as

• You perform a function on x, in this case you square it to get y.

• So f(4)=16, f(-4)=16, f(3)=9, etc.

• Notice no x’s are repeated , so this is a function.

• The x-value, which can vary, is called the independent variable, and the y-value, which is determined from “doing something” to x is called the dependent variable

• Functions can be represented in words: (square x to get y)

• in a table of values:

• in function notation:

• Or on a graph:

Note*

• We use “function notation” to substitute an x-value into an equation and find its y-value

Examples

1. For , find:

a) f(-3)b) f(21)c) f(w+4)d) 3f(5)

Assignment• Ex. 2.1 (p. 55) #1-10

2. Identifying Functions

a) Polynomial Functions

• n is a nonnegative integer and , , etc. are coefficients

Example

1. For , find the leading coefficient and the degree.

• The polynomials function has degree “n” (the largest power) and leading coefficient

Example

2. For , find the leading coefficient and the degree.

• A polynomial function of degree 0 are called constant functions and can be written f(x)=b

• Slope = zero

Example

1.

• Polynomial functions of degree 1 are called linear functions and can be written

• y = mx + b

• m= slope• b= y-intercept

• Example Graph

• The linear function is also called the identity function

• Example graph

• Polynomial functions of a degree 2 are called quadratic functions and can we written

• Example graph

• Polynomial functions of degree 3 are called cubic functions

• Example Graph

• Degree 4 functions with a negative leading coefficient

• Example Graph

• Degree 5 functions with a negative leading coefficient

• Example graph

• Summary: • Polynomial functions of an odd degree and positive leading

coefficient begin in quadrant 3 and end in quadrant 1

• Polynomial functions of an odd degree and negative leading coefficient begin in quadrant 2 and end in quadrant 4

• Polynomial functions of an even degree and positive leading coefficient begin in quadrant 2 and end in quadrant 1

• Polynomial functions of an even degree and negative leading coefficient begin in quadrant 3 and end in quadrant 4

b) A Power Function can be written:

• where n is a real number

• If “n” is a positive integer, the power function is also a polynomial function

• Examples

Examples

1. Graph the following on your graphing calculator:

etc.

• Notice that all the graphs pass through the points (0,0) and (1,1).

• This is true for all power functions

• If the power is and n is a positive integer >1, it is called a root function

Graph the following and find the interval for each.

• If the power is negative, it is called a reciprocal function and can be written:

• Its graph is an hyperbola with x and y axes as asymptotes.

• Example Graph

c) A Rational Function is the ratio of 2 polynomial functions and can be written:

Note*: the reciprocal function is also a rational function.

• Any x-value which makes the denominator = 0 is a vertical asymptote.

• If degree of p(x) < degree of q(x), there is a horizontal asymptote at y=0 (x-axis)

• If degree of p(x) = degree of q(x), there is a horizontal asymptote at y = k, where k is the ratio of the leading coefficients of p(x) and q(x) respectively.

Example

1. Find the asymptotes of the following function.

d) An Algebraic Function is formed by performing a finite number of algebraic operations (such as with polynomials

• Thus all rational functions are also algebraic functions.

Examples

Using your graphing calculators graph the following:

• Thus the graphs of algebraic functions vary widely.

e) Trig Functions – contain sin, cos, tan, csc, sec or cot.

Examples

f) Exponential Functions have “x” as the exponent (rather than as the base, as in power functions) and can be written:

• where b>0,

• Graphs of exponential functions always pass through (0,1) and lie entirely in quadrants 1 and 2

• If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The x axis is a horizontal asymptote line.

Example• Graph the following.

g) Logarithmic Functions have “y” as the exponent and can be written

• where b>0,

• Graphs of log functions always pass through (1,0) and lie in quadrants 1 and 4

• If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The y-axis is a vertical asymptote line.

Examples• Graph the following.

h) Transcendental Functions are functions that are not algebraic. They included the trig functions, exponential functions, and the log functions

Assignment• Ex. 2.2 (p. 64) #1-4

3. Piecewise and Step Function

a) A Piecewise Function is one that uses different function rules for different parts of the domain.

• Watch open and closed intervals and use corresponding dots

• To find values for the function, use the equation that contains that value (on the graph) in its domain.

Example

1. Graph

2. Find:a) f(-11)b) f(7)c) f(0)

• The Absolute Value Function is a piecework-defined function:

• Graph

b) The graph of a step function looks like a series of steps.

• The greatest integer function names the greatest integer that is less than or equal to x and is written

Examples

• This function is also called the floor function because the function rounds non-integer values down.

• The notation, is also used

Example• Graph

• Using your graphing Calculator

• There is also a function which returns the smallest integer that is greater than or equal to x,

Examples

• In other words, this function rounds non-integer values up and is called the least integer function or ceiling function

Assignment• Ex. 2.3 (p. 70) #1-5

4. Characteristics of Functions

a) A function is said to be even if it is symmetrical around the y-axis.

• That is, f(x) and f(-x) are the same value

Examples• Graph the following using your graphing calculators.

• Notice that every point (a,b) is the 1st quadrant has a mirror image, (-a,b) in the second quadrant

b) A function is said to be odd, if it is symmetrical around the origin.

• That is,

Examples• Graph the following using your calculator.

• Notice that every point (a,b) has a corresponding point (-a, -b)

• Can a function be both even and odd? Explain/Prove.

c) A function is increasing if it rises from left to right and decreasing if it falls from left to right

• A function is increasing on an interval I if whenever in the interval I.

• A function is decreasing on an interval I if whenever in the interval I.

Example• Determine if the function is increasing or decreasing and on

what intervals.

1. g

d) A function is one-to-one if neither the x nor the y-values are repeated

• Examples

A function is many-to-one if y-values are repeated

• Examples

• What is mapping notation?

• Can a function be one-to-many? Why or why not?

Assignment• Ex. 2.4 (p. 79) #1-9

5. Graphing Transformations

a) Vertical Shirts – simply add “c” to shift up “c” units and subtract “c” to shift graph down “c” units

Examples• Graph the Following • Graph the Following

4.

5.

6.

b) Horizontal Shifts – for f(x), f(x+c) will shift the graph “c” units to the left and f(x-c) will shift the graph “c” units to the right

Examples• Graph the Following • Graph the Following

4.

5.

6.

c) Vertical Stretches – for f(x), c(f(x)) where c>1, will stretch the graph vertically by “c” units

• That is, all the y-values are “c” times higher than before (multiply the y by c)

Examples• Graph the Following • Graph the Following

4.

5.

6.

d) Vertical Compressions for f(x), , where c>1, will compress the graph vertically by c units

• That is, all the y-values are times as high as the were before (divide y by c)

Examples• Graph the Following • Graph the Following

4.

5.

6.

e) Horizontal Compressions – for f(x), f(cx), where c>1, will compress the graph horizontally by c units.

• That is, the function reaches its former y-values c times sooner. (divide x by c)

Examples• Graph the Following • Graph the Following

4.

5.

6.

f) Horizontal Stretches – for f(x), where c>1, will stretch the graph horizontally by c units.

• That is, the function reaches its former y-values c times later (multiply x by c)

Examples• Graph the Following • Graph the Following

4.

5.

6.

g) Reflection about the x-axis: to reflect a function such as around the x-axis, simply enter

• y becomes –y

Examples• Graph the left then

the right:

i) ii) iii) iv) v)

i) ii) iii) iv) v)

h) Reflection about the y-axis: to reflect a function such as around the y-axis, simply enter

Example• Graph the left then the right:

• How should we transform to obtain the graphs of the following:

Assignment• Ex. 2.5 (p. 90) Oral Ex. 1-15 Written 1-36 odds

6. Finding Domain and Range

a) The Domain (x-values) and Range (y-value) may be determined b examining the graph of the function

Examples

Graph the following to find the domain and range.

b) The domain and range of the function can also be determined by examining the equation of a function.

• You analyze the equation for restrictions on the domain. That is, are there any x-values that would make a denominator equal to zero or a negative value under an even root sign.

• Generally, restrictions on the domain will cause restrictions of the range.

Example• Find the domain and range of the following equaitons.

Examples• Find the domain and range of the following equaitons.

5.

6.

7.

8.

• Recall that you cannot find the logarithm for a non-positive number

Example• Find the domain and range of the following equaitons.

8.

9.

10.

11.

• Domain Summary

• You cannot divide by zero.

• You cannot take the even root of a negative number.

• You cannot find the logarithm of a non-positive number.

• Finding The Range • There is no rule for finding the range of a function. Generally

students need to be asking themselves questions such as:

• What happens to the value of the function for large positive x values? • What happens to the value of the function for large negative x values? • What happens to the value of the function near to any values in the domain that cause the denominator of the function to be zero? • Do the numerator, denominator, or any part of the expression ever reach a minimum/maximum value? • Determining the horizontal and vertical asymptote lines (Math B30) together with a sign analysis is helpful for rational functions.

Assignment• Ex. 2.6 (p. 99) #1-45 odds

7. Combinations of Functions

a) Functions can be combined using .

Example• For

1. Find

For , the domain of f(x) is [2,∞) and the domain of g(x) is (- ∞,6). Therefore, the domain of can also be written is [2,∞)

• The domain of is the intersection of their 2 domains.

• The same is true for the domain of , and provided

Example• What is the domain of

b) You can also take a “function of a function”

• Remember to start from the inside brackets and work out.

• can also be written

• The domain of is the set of all values in the domain of g such that g(x) is in the domain of f.

• The domain of is the set of all values in the domain of f such that f(x) is in the domain of g.

Examples• For , find:

1. f(2)2. g(2)

Assignment• Ex. 2.7 (p. 106) #2-16