b. functions calculus 30. 1. introduction a relation is simply a set of ordered pairs. a function is...
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B. FunctionsCalculus 30
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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.
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• 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.
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• 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:
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Note*
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• We use “function notation” to substitute an x-value into an equation and find its y-value
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Examples
1. For , find:
a) f(-3)b) f(21)c) f(w+4)d) 3f(5)
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Assignment• Ex. 2.1 (p. 55) #1-10
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2. Identifying Functions
a) Polynomial Functions
• n is a nonnegative integer and , , etc. are coefficients
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Example
1. For , find the leading coefficient and the degree.
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• The polynomials function has degree “n” (the largest power) and leading coefficient
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Example
2. For , find the leading coefficient and the degree.
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• A polynomial function of degree 0 are called constant functions and can be written f(x)=b
• Slope = zero
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Example
1.
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• Polynomial functions of degree 1 are called linear functions and can be written
• y = mx + b
• m= slope• b= y-intercept
• Example Graph
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• The linear function is also called the identity function
• Example graph
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• Polynomial functions of a degree 2 are called quadratic functions and can we written
• Example graph
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• 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
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• 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
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b) A Power Function can be written:
• where n is a real number
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• If “n” is a positive integer, the power function is also a polynomial function
• Examples
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Examples
1. Graph the following on your graphing calculator:
etc.
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• Notice that all the graphs pass through the points (0,0) and (1,1).
• This is true for all power functions
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• If the power is and n is a positive integer >1, it is called a root function
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Graph the following and find the interval for each.
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• 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
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c) A Rational Function is the ratio of 2 polynomial functions and can be written:
Note*: the reciprocal function is also a rational function.
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• 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.
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Example
1. Find the asymptotes of the following function.
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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.
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Examples
Using your graphing calculators graph the following:
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• Thus the graphs of algebraic functions vary widely.
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e) Trig Functions – contain sin, cos, tan, csc, sec or cot.
Examples
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f) Exponential Functions have “x” as the exponent (rather than as the base, as in power functions) and can be written:
• where b>0,
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• 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.
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Example• Graph the following.
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g) Logarithmic Functions have “y” as the exponent and can be written
• where b>0,
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• 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.
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Examples• Graph the following.
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h) Transcendental Functions are functions that are not algebraic. They included the trig functions, exponential functions, and the log functions
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Assignment• Ex. 2.2 (p. 64) #1-4
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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.
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Example
1. Graph
2. Find:a) f(-11)b) f(7)c) f(0)
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• The Absolute Value Function is a piecework-defined function:
• Graph
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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
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Examples
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• This function is also called the floor function because the function rounds non-integer values down.
• The notation, is also used
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Example• Graph
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• Using your graphing Calculator
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• There is also a function which returns the smallest integer that is greater than or equal to x,
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Examples
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• In other words, this function rounds non-integer values up and is called the least integer function or ceiling function
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Assignment• Ex. 2.3 (p. 70) #1-5
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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
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Examples• Graph the following using your graphing calculators.
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• Notice that every point (a,b) is the 1st quadrant has a mirror image, (-a,b) in the second quadrant
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b) A function is said to be odd, if it is symmetrical around the origin.
• That is,
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Examples• Graph the following using your calculator.
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• Notice that every point (a,b) has a corresponding point (-a, -b)
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• Can a function be both even and odd? Explain/Prove.
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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.
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Example• Determine if the function is increasing or decreasing and on
what intervals.
1. g
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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
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• What is mapping notation?
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• Can a function be one-to-many? Why or why not?
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Assignment• Ex. 2.4 (p. 79) #1-9
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5. Graphing Transformations
a) Vertical Shirts – simply add “c” to shift up “c” units and subtract “c” to shift graph down “c” units
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Examples• Graph the Following • Graph the Following
4.
5.
6.
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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
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Examples• Graph the Following • Graph the Following
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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)
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Examples• Graph the Following • Graph the Following
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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)
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Examples• Graph the Following • Graph the Following
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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)
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Examples• Graph the Following • Graph the Following
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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)
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Examples• Graph the Following • Graph the Following
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g) Reflection about the x-axis: to reflect a function such as around the x-axis, simply enter
• y becomes –y
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Examples• Graph the left then
the right:
i) ii) iii) iv) v)
i) ii) iii) iv) v)
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h) Reflection about the y-axis: to reflect a function such as around the y-axis, simply enter
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Example• Graph the left then the right:
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• How should we transform to obtain the graphs of the following:
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Assignment• Ex. 2.5 (p. 90) Oral Ex. 1-15 Written 1-36 odds
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6. Finding Domain and Range
a) The Domain (x-values) and Range (y-value) may be determined b examining the graph of the function
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Examples
Graph the following to find the domain and range.
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b) The domain and range of the function can also be determined by examining the equation of a function.
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• 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.
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Example• Find the domain and range of the following equaitons.
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Examples• Find the domain and range of the following equaitons.
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• Recall that you cannot find the logarithm for a non-positive number
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Example• Find the domain and range of the following equaitons.
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• 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.
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• 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.
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Assignment• Ex. 2.6 (p. 99) #1-45 odds
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7. Combinations of Functions
a) Functions can be combined using .
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Example• For
1. Find
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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,∞)
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• The domain of is the intersection of their 2 domains.
• The same is true for the domain of , and provided
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Example• What is the domain of
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b) You can also take a “function of a function”
• Remember to start from the inside brackets and work out.
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• 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.
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Examples• For , find:
1. f(2)2. g(2)
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Assignment• Ex. 2.7 (p. 106) #2-16