Functions Definition and Notation Domain and Range Examples How to find the domain Evaluation Examples and arithmetic reminders Graphing Intercepts and asymptotes Summary of Basic Functions Polynomials Rational Functions Special Functions Sequences Notation Domain and Range Bounded Sequences Appendices Interval Notation Factoring The Guaranteed Method Complete the Square Method Factoring by Grouping Homework on my website 1 Definition and Notation A function is a table chart graph or rule that associates one element from a set called the domain with exactly one element from a set called the range When a computational rule is given the function is usually denoted f x Example 1 Given Using two finite sets D 0 1 2 R 1 3 5 One function that can be created from these two sets is 0 1 1 3 2 5 Graph Another is 0 5 1 1 3 2 Graph The cross product of these two sets is NOT a function because a domain element is associated with 3 range elements not exactly one 0 1 0 3 0 5 1 1 1 3 1 5 2 1 2 3 2 5 Graph reminder Vertical Line Test Each function that can be created is a SUBSET of the cross product though 2 Example 2 A linear function f x 3x 1 For this function the domain is all real numbers and so is the range We use interval notation to indicate this Domain Range Graph for a review of interval notation see the handout in the Functions folder on my website x intercept and y intercept 0 0 3 Example 3 Restricted Domains and Ranges Sometimes it happens that some real numbers are not allowed in the calculation of point pairs this creates domain restrictions And sometimes not every real number is calculated as a second coordinate which makes range restrictions 3 A A restricted range f x x 2 4x 5 Domain Range 9 You do not use every real number on the y axis as a second coordinate No matter which x you use you will get a y that is greater than or equal to 9 Factoring handout on my website x intercepts y intercept Turn around point Complete the Square handout on my website 4 3 B A restricted domain and range f x x 3 Domain 3 Range 0 You cannot take the square root of a negative number and get a real number you get a complex number which doesn t graph in the Cartesian Plane So since we re interested in graphs we say the domain is restricted to only those numbers that will produce a graphable y Note that you can see the domain and range of the function on the graph 5 3 C f x A restricted domain and range 2x 3 x 1 Domain 1 1 Range 2 2 Vertical Asymptote Horizontal Asymptote 6 How to find the domain The domain of a function is ALWAYS all real numbers unless the function has one of the following problems a square root a divide by an expression with an x a log as a function name we won t be doing any of these We will use the graph to find the range unless the function is a very basic one Square roots n b or any even root We all know you can t take the square root of a negative number and get a real number you get an imaginary number that you can t graph in the Cartesian Plane So if you re working with point pairs and you ve got an x that gives you an imaginary y you throw it out and say it s not in the domain Doing this x by x takes too long So we do it with algebra You CAN take the square root of zero and any positive number So Take the expression under the radical Set it greater than or equal to zero Solve for x Report the solution in interval notation 7 EXAMPLE 1 What is the domain for Take the expression under the radical Set it greater than or equal to zero 2x 12 2 x 12 0 2 x 16 Solve for x x 6 6 Report the solution in interval notation EXAMPLE 2 f x 2 x 12 What is the domain for f x 15 3x 15 3x 0 3x 15 x 5 5 EXERCISE 1 What s the domain for f x 5x 10 Take the expression under the radical Set it greater than or equal to zero Solve for x Report the solution in interval notation Divides with an expression that contains an x note that dividing by a number like 3 is not a domain problem 8 the domain for f x 2x 2 1 is all Real Numbers 5 We all know you can t divide by zero So Take the expression in the denominator Set it equal to zero Solve for x Throw them out off the number line Report THE REST as the domain EXAMPLE 3 Find the domain for f x Take the expression in the denominator Set it equal to zero x2 1 x 2 5x 6 x 2 5 x 6 0 x 2 x 3 0 x 2 or x 3 Solve for x Throw them out off the number line 2 3 Report THE REST as the domain 2 2 3 3 EXAMPLE 4 Find the domain for f x 4 x 9 2 x 2 9 0 x 3 x 3 0 x 3 or x 3 9 3 3 3 3 3 3 EXERCISE 2 Find the domain for f x this is the domain 12 x 3x 2 2 Take the expression in the denominator Set it equal to zero Solve for x Throw them out off the number line Report THE REST as the domain Summary Tell me the domains for 10 f x x 4 f x x 4 2 f x x 4 f x x 4 f x 1 x 4 f x x 4 3 f x 1 x 5x 4 2 f x 4 x 1 11 Evaluation An excursion through minus signs fractions and exponents Given Evaluate f x x 2 x 2 f 1 1 2 1 2 4 So 1 4 is a graph point of the given function Replace each x with the indicated value and compute the y f x Notes 1 x remember x 1 remember 32 9 remember a fractional exponent is root finding vs 3 2 9 1 f x x 2 x 1 f x x 3 3 x 12 EXAMPLE 1 f x x 2 What is the domain for this function f 2 2 2 1 4 1 Calculate f 2 EXAMPLE 2 f x x 1 x 2 What is the domain for this function Calculate f 1 f 0 f 1 2 13 EXAMPLE 3 1 f x x 2 Calculate f 0 f 4 f 2 Graph What are the domain and range for this graph 14 Example 4 Given two points let s review how to get the formula for the line they determine …