Real-Valued Functions A Real-Valued Function f of a real variable x from set X to set Y is a correspondence that assigns to each number x in set X exactly one number y in set Y X Domain x Y f Range Y = f(x) X and Y are sets of real numbers This sounds familiar but the words seem different. The domain of f is the set X. The number y is the image of x under f and is denoted by f (x). The range of f is a subset of Y and consists of all images of numbers in set X

Implicit vs Explicit Form Defines y, the dependent variable, as a function of x, the independent variable. Implicit Form Defines the equation in terms of y. Use the Explicit Form to evaluate the function. Explicit Form f(x) and y are interchangeable. Function Notation

Evaluating Functions Evaluate each of the following.

Evaluating Functions Evaluate each of the following.

More Evaluating Functions Evaluate for This looks hard, but it really isn’t that bad.

Evaluating Functions Worksheet Homework Evaluating Functions Worksheet

Homework Problem 1 Asi de Facil Evaluate for Holy Schnikies! That’s really pretty darn easy. Asi de Facil

Homework Problem 2 Evaluate for That was easy

Homework Problem 3 Evaluate for That was easy

Evaluate for

Homework Evaluating Functions Worksheet #2

Intervals on the Real Line Set Notation Interval Notation Graph ( ) Bounded Open Interval (a, b) a b x [ ] Bounded Closed Interval [a, b] a b x [ ) [a, b) Bounded Intervals (neither open or closed) a b x ( ] (a, b] a b x ) Unbounded Open Intervals a b x ( a b x ] Unbounded Closed Intervals a b x [ x a b Entire Real Line a b x

Domain and Range of a Function The domain of a function can be described implicitly, or the domain can be described explicitly. This equation has an implicitly defined domain that is the set This equation has an explicitly defined domain given by An implied domain is the set of all real numbers for which the equation is defined. An explicitly defined domain is one that is given along with the function.

Finding the Domain and Range of a Function Find the domain and range of The domain is the set of all x-values for which That means the domain is the interval Since f(x) is never negative, that means the range is the interval Let’s look at a diagram.

More Domain and Range of a Function Find the domain and range of each of the following functions. If both polynomials are the same degree, divide the coefficients of the highest degree terms. If the numerator is lower degree than the denominator, y = 0 is the asymptote. There is a horizontal asymptote at y = 1 If the numerator is higher degree than the denominator, there is no asymptote. Asi de Facil

Find Domain & Range for all functions Homework Page 27: 13 – 16, 19, 20, 25, 26 Find Domain & Range for all functions

Functions Defined by More than One Equation Find the domain and range of Let’s look at a diagram. This is called a Piecewise Function. That was easy

Piecewise Functions 1 That was easy Evaluate the function as indicated. Determine the domain and range. That was easy

Piecewise Functions b That was easy Evaluate the function as indicated. Determine the domain and range. That was easy

Piecewise Functions 3 That was easy Evaluate the function as indicated. Determine the domain and range. That was easy

Piecewise Functions d That was easy Evaluate the function as indicated. Determine the domain and range. That was easy

Piecewise Functions 5 That was easy Evaluate the function as indicated. Determine the domain and range. That was easy

Homework Page 27: 28, 29, 30 All Parts

Graphs of Functions The graph of a function y = f(x) consists of all points (x, f(x) ), where x is in the domain of f. x = the directed distance from the y-axis. f(x) = the directed distance from the x-axis. f(x) x

Graphs of Basic Functions Identity Function Squaring Function Cubing Function Square Root Function

More Graphs of Basic Functions Absolute Value Function Rational Function Sine Function Cosine Function

Vertical Line Test A graph represents a function if any vertical line drawn intersects it in at most one point. The vertical line never intersects the graph in more than one point, therefore this is a function. The vertical line never intersects the graph in more than one point, therefore this is a function. The vertical line does intersect the graph in more than one point, therefore this is not a function.

Transformations of Functions Let’s take a look at the graph of Original function Horizontal shift c units to the right Horizontal shift c units to the left Vertical shift c units downward Vertical shift c units upward Reflection (about the x-axis) Reflection (about the y-axis) Reflection (about the origin)

Transformation Example Use the graph of f shown in the figure to sketch the graph of each function.

Homework Page 28: 41 – 44, 55, 56

I didn’t realize that this was that easy. Polynomial Functions n is the degree of the function, the numbers ai are the coefficients an is the leading coefficient, and a0 is the constant term Let’s look at a sample with some actual numbers. 4 is the degree of the function, 3 is the leading coefficient, and -7 is the constant term 5is the degree of the function, -6 is the leading coefficient, and 3 is the constant term I didn’t realize that this was that easy. That was easy

Turning Points of Functions The number of turning points that a polynomial function has can be determined by the degree of the equation. The number of turning points of a polynomial function is equal to n - 1 One turning point Two turning points Four turning points

End Behavior of Functions There are two factors that determine the End Behavior of a function. End Behavior of Functions Degree of the Function Even Polynomials of even degree have ends going in the same direction. Polynomials of odd degree have ends going in opposite directions. Odd Sign of the Leading Coefficient Even Odd Even Odd Up to left Up to right Down to left Up to right Down to left Down to right Up to left Down to right That’s a little confusing but pretty easy.

Describing the Behavior of a Function For each of the following functions a) state the number of turning points b) describe the end behavior Positive Even 3 turning points Up to left Up to right Negative Odd Up to left Down to right 4 turning points Negative Even Negative Even 5 turning points Down to left Down to right Asi de Facil 3 turning points Down to left Down to right

Function Behavior Worksheet Homework Function Behavior Worksheet

Zeros of a Function The zeros of a function are the solutions of the equation f(x)=0 Let’s take a look at some examples Example 1 Example b Since f(0) = 0, f(-1) = 0 and f(1) = 0 the zeros of the function are: x = 0, x = -1, x = 1 Since f(4) = 0, x = 4 is a zero of the function.

Even and Odd Functions An Even Function is symmetric with respect to the y-axis. Test for Even and Odd Functions An Odd Function is symmetric with respect to the origin. Example 1 Example b Example 3 The Function is Even The Function is Odd The Function is Odd

More Even and Odd Functions Find the coordinates of a second point on the graph of a function if the given point is on the graph and the function is (a) even and (b) odd. That was easy Remember: Example 1 Example b Example 3 (-1, -4) (3, 7) (-6, 2) (1, -4) even (-3, 7) (6, 2) (1, 4) odd (-3, -7) (6, -2)

Practice Examples for Even and Odd Functions Example b Example 3 Note: Even Even Example d Odd Note: Even

Homework Page 29: 69 - 75