Definition, Domain,Range,Even,Odd,Trig,Inverse Function Definition, Domain,Range,Even,Odd,Trig,Inverse Dr. Hajjaj--Lecture-1-Date-2-10

Common functions sine, cosine, tangent (and their inverses) logarithm, exponential sinh and cosh straightline, quadratic, general polynomial combinations of above as products, composites and fractions. You should be familiar with shapes of common functions and be able to sketch quickly.

Dr. Hajjaj--Lectur-1-Date11-10 Function Function - for every x there is exactly one y. Domain - set of x-values Range - set of y-values Dr. Hajjaj--Lectur-1-Date11-10

Tell whether the equations represent y as a function of x. a. x2 + y = 1 Solve for y. y = 1 – x2 For every number we plug in for x, do we get more than one y out? No, so this equation is a function. Solve for y. b. -x + y2 = 1 y2 = x + 1 Here we have 2 y’s for each x that we plug in. Therefore, this equation is not a function.

Find the domain of each function. a. f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)} Domain = { -3, -1, 0, 2, 4} b. D: c. D: [-2, 2]

(2,4) Find: the domain the range f(-1) = f(2) = [-5,4] (4,0) [-1,4) -5 (-1,-5) Dr. Hajjaj--Lecture-1

Vertical Line Test for Functions Do the graphs represent y as a function of x? yes yes no Dr. Hajjaj--Lecture-1

Summary of Graphs of Common Functions—see page 60 in textbook f(x) = c y = x y = x 3 y = x2 Dr. Hajjaj--Lecture-1

What is a function? A rule which translates an input, usually to a single output. What are the functions for: Double the input Shift the input by 3 Cube the input and subtract 1. Write down in words the functions for Dr. Hajjaj--Lecture-1

What variables can a function have? What is the difference between the functions f(x), g(w), h(y) and k(x) A function describes a relationship, the variable names are unimportant. Engineers typically use variable names that relate to the topic: W for weight, h for height, L for length, etc. Dr. Hajjaj--Lecture-1-

What is a function argument? The part that appears in the brackets; For y=f(x), x is the argument. For z=g(w), w is the argument. Thus argument is another word for the input to the function. Independent and dependent variables: what do you think these are? Use common sense. Dr. Hajjaj--Lecture-1

Example [Fluid Mechanics] The streamlines of fluid flow are given by: y = x2 + c where c is constant. Sketch the streamlines for c = 0, -1 ,1, -2, 2, -3 and 3. Dr. Hajjaj--Lecture-1

Example Solution The graphs of y = x2 + c for c = 0, -1 ,1, -2, 2, -3 and 3 are: (c=0) y = x2 (c=-1) y = x2 - 1 (c=1) y = x2 + 1 (c=-2) y = x2 -2 (c=2) y = x2 +2 (c=-3) y = x2 – 3 (c=3) y = x2 + 3 Dr. Hajjaj--Lecture-1

y c = 3 c = 2 c = 1 c = 0 c = -1 3 c = -2 2 c = -3 1 -2 -1 1 2 x -1 -2 -3 Notice how the graph of y=x2 + c varies as c changes. The c is where the curve cuts the y axis. Dr. Hajjaj--Lecture-1

Types of Functions n is a nonnegative integer, each is a constant. Polynomial Functions n is a nonnegative integer, each is a constant. Ex. Rational Functions polynomials Ex. Dr. Hajjaj--Lecture-1-

Types of Functions Power Functions ( r is any real number) Ex. Ex. Dr. Hajjaj--Lecture-1-

FUNCTIONS Symmetric about the y axis Symmetric about the origin Dr. Hajjaj--Lecture-1

Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Dr. Hajjaj--Lecture-1

Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Dr. Hajjaj--Lecture-1

x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3 -4 -5 -6 -7 Dr. Hajjaj--Lecture-1

A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO Dr. Hajjaj--Lecture-1

A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES Dr. Hajjaj--Lecture-1-

Determine if the following functions are even, odd or neither. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN. Dr. Hajjaj--Lecture-1-

composition of Functions The composition of the functions f and g is “f composed by g of x equals f of g of x” Dr. Hajjaj--Lecture-1

Ex. f(x) = x + 2 and g(x) = 4 – x2 Find: of 2 Ex. f(x) = x + 2 and g(x) = 4 – x2 Find: f(g(x)) = (4 – x2) + 2 = -x2 + 6 g(f(x)) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4) Dr. Hajjaj--Lecture-1 = -x2 – 4x

Ex. Express h(x) = as a composition of two functions f and g. f(x) = g(x) = x - 2 Dr. Hajjaj--Lecture-1

Inverse Functions Dr. Hajjaj--Lecture-1

Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) Ex: and are inverses because their input and output are switched. For instance: 22 4 Dr. Hajjaj--Lecture-1 22 4

Inverse and Compositions In order for two functions to be inverses: AND Dr. Hajjaj--Lecture-1

One-to-One Functions A function f(x) is one-to-one on a domain D if, for every value c, the equation f(x) = c has at most one solution for every x in D. Or, for every a and b in D: Theorems: A function has an inverse function if and only if it is one-to-one. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. Dr. Hajjaj--Lecture-1

The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions. Dr. Hajjaj--Lecture-1

Remember we talked about functions---taking a set X and mapping into a Set Y 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5 Set X Set Y An inverse function would reverse that process and map from SetY back into Set X Dr. Hajjaj--Lecture-1-

The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions. Dr. Hajjaj--Lecture-1

This would not be a one-to-one function because to be one-to-one, each y would only be used once with an x. 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 5 10 This is a function IS one-to-one. Each x is paired with only one y and each y is paired with only one x Only one-to-one functions will have inverse functions, meaning the mapping back to the original values is also a function. Dr. Hajjaj--Lecture-1

Verify that the functions f and g are inverses of each other. If we graph (x - 2)2 it is a parabola shifted right 2. Is this a one-to-one function? This would not be one-to-one but they restricted the domain and are only taking the function where x is greater than or equal to 2 so we will have a one-to-one function. Dr. Hajjaj--Lecture-1

Verify that the functions f and g are inverses of each other. Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses. Dr. Hajjaj--Lecture-1

Steps for Finding the Inverse of a One-to-One Function y = f -1(x) Solve for y Trade x and y places Replace f(x) with y Dr. Hajjaj--Lecture-1

Let’s check this by doing Find the inverse of y = f -1(x) or Solve for y Trade x and y places Yes! Replace f(x) with y Ensure f(x) is one to one first. Domain may need to be restricted. Dr. Hajjaj--Lecture-1

Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 2. The range is the set of y values such that . 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 5. Each function cycles through all the values of the range over an x-interval of . 6. The cycle repeats itself indefinitely in both directions of the x-axis. Dr. Hajjaj--Lecture-1

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1 sin x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x Dr. Hajjaj--Lecture-1 Sine Function

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1 cos x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x Dr. Hajjaj--Lecture-1 Cosine Function

Example: Sketch the graph of y = 3 cos x on the interval [–, 4]. Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Dr. Hajjaj--Lecture-1 Example: y = 3 cos x

Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x y = sin (–x) Use the identity sin (–x) = – sin x y = sin x Example 2: Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. y x Use the identity cos (–x) = – cos x y = cos (–x) y = cos (–x) Dr. Hajjaj--Lecture-1 Graph y = f(-x)

Graph of the Tangent Function To graph y = tan x, use the identity . At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y x period: Dr. Hajjaj--Lecture-1 Tangent Function

Graph of the Cotangent Function To graph y = cot x, use the identity . At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y x vertical asymptotes Dr. Hajjaj--Lecture-1 Cotangent Function

Graph of the Secant Function The graph y = sec x, use the identity . At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y x Dr. Hajjaj--Lecture-1 Secant Function

Graph of the Cosecant Function To graph y = csc x, use the identity . At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. x y Dr. Hajjaj--Lecture-1 Cosecant Function