Functions and Their Graphs Section P.3

After this lesson, you should be able to: evaluate a function recognize the graphs of linear, squaring, cubing, square root, absolute value, rational, sine and cosine functions and graph them on a calculator find the domain and range of a function use the graphing calculator to graph piecewise functions describe transformations of the basic functions find the zeroes of a function determine if a function is even or odd

Definition of Function Real-Valued Function of a Real Variable: Let X and Y be sets of real numbers. A real-valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X exactly one number y in Y. The domain of f is the set X. is the image of x under f and is denoted by f(x). The number y The range of f is a subset of Y and consists of all images of numbers in X.

Functions X Y f domain range x y

Common Forms for Describing Functions MAT 224 SPRING 2007 Common Forms for Describing Functions Typically, x is the independent variable and y is the dependent variable. Implicit form: Explicit form (Solve for y in terms of x): Function notation:

Example—Evaluating a Function MAT 224 SPRING 2007 Example—Evaluating a Function For the function f defined by , evaluate each expression. a) b)

Example—Evaluating a Function (cont.) MAT 224 SPRING 2007 Example—Evaluating a Function (cont.) For the function f defined by , evaluate each expression. c)

Domain and Range of a function More simply stated, we can say Domain x-values (or input values) Range y-values (or output values) Try thinking of a function as a machine with a “crank”. The domain values are those you put into the machine. Turn the crank and the range values are those that come out the machine. So think about putting in domain values for x and getting out the corresponding range values for y (or f(x)). You’ve done this when you created tables of values.

Domain of a Function The domain of a function may be described explicitly. Example: The domain may also be described implicitly. The implied domain is the set of all real numbers for which the function is defined. Example: Define the domain of the function:

Interval Notation-Refresher Open interval: (a, b) Closed interval: [a, b] (a, b] [a, b) Note: The interval is ALWAYS open where infinity is part of the interval notation.

Domain and Range of a Function Example: Graph the function on your calculator, and then find the domain and range of the function.

Domain and Range--Piecewise Function Example: Graph on your calculator and then determine the domain and range of the function. You can find the inequality symbols using   to get the TEST menu

Evaluate a piecewise function Example: For the piecewise function defined, evaluate each expression.

Vertical Line Test To determine if a graph is a function, a vertical line test can be performed. In order for a graph to be a function, a vertical line in any spot on the graph can only intersect the graph at most once.

Basic Functions The graphs of the eight basic functions from page 22 in your text should be recognizable to you from now on. Absolute Value Function: Identity Function: Rational Function: Squaring Function: Cubing Function: Sine Function: Square Root Function: Cosine Function:

Graphs of the Basic Functions (pt 1) Identity Function Squaring Function Domain: Range:

Graphs of the Basic Functions (pt 2) Cubing Function Square Root Function Domain: Range:

Graphs of the Basic Functions (pt 3) Absolute Value Function Rational Function Domain: Range:

Graphs of the Basic Functions (pt 4) Sine Function Cosine Function Domain: Range:

One-To-One Function A function from X to Y is one-to-one if to each y-value in the range, there corresponds exactly one x-value in the domain. Passes Horizontal Line Test

One-To-One Function The following functions are NOT one-to-one. These graphs do NOT pass the Horizontal Line Test

Transformations of Functions Let c > 0. y = f(x) Original graph y = -f(x) y = f(-x) y = f(x – c) y = f(x + c) y = f(x) + c y = f(x) - c

Transformations of Functions Describe the transformations that the graph of f(x) must undergo to obtain the graph of g(x). Orig:

Polynomial Functions The most common algebraic function is the polynomial function. n is a positive integer (the degree of the polynomial function) ai are coefficients (an is the leading coefficient and a0 is the constant term)

Rational Functions The function f(x) is rational if Polynomial, rational, and radical functions are all algebraic functions. The other types are trig and exponential functions. Non-algebraic functions are called transcendental functions.

Elementary Functions Elementary Functions Types Algebraic Trigonometric Exponential & Logarithmic Examples Polynomial Radical Rational Sine Cosine Tangent Cotangent Secant Cosecant

Composite Functions Let f and g be functions. The function given by (f ° g)(x) = f(g(x)) is called the composite of f with g. The domain of f ° g is the set of all x in the domain of g such that g(x) is in the domain of f. x f(g(x)) f ° g g f g(x)

Example 1-Composite Functions *Example: Given f(x) = x2 – 1 and g(x) = cos x, find f ° g. (Pythagorean Identity cos2x+ sin2x = 1)

Example 2-Composite Functions Example: Given and , find

Tests for Even and Odd Functions The function y = f(x) is even if f(-x) = f(x) (has y-axis symmetry) The function y = f(x) is odd if f(-x) = -f(x) (has origin symmetry)

Example of Odd and Even functions Example Determine whether the function is even, odd, or neither. a) b)

Homework Section P.3: page 27 #1, 3, 7, 17, 19, 23-27 odd, 29-41 odd, 55, 59, 61, 93 (a and b only)