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

Homework On HW Worksheet Page of Website 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

On HW Worksheet Page of Website Homework On HW Worksheet Page of Website Page 27: 28, 29, 30 All Parts