Precalculus – Hollyer Chapter 1: Functions1This course is all about functions. So let’s start with a definition of the word “function”.We need 2 sets (S1 and S2) and a rule that assigns each element in S1 to one and only oneelement in S2. S1 is called the domain of the function and S2 is called the range of the function.On my website, there is a handout on “Basic Graphs” that will help you review all the functions you’ll need for this course and for the Calculus sequence.We use functional notation to give the rule. We start with the math expression “f(x)” and set it equal to another, computational, math expression. Sometimes, in a long or complicated problem other letters will be used in addition to “f” to indicate that the expression is a function (g, p, h, and l are commonly used instead of “f”).For our purposes, both the domain and range are subsets of the real numbers. You will beexpected to state the domain and range of a function in interval notation. You will be expected to be able to figure out the domain of a function analytically and to find the range of the function from the graph of the function.Unless you have an identifiable problem, the domain is all real numbers (in interval notation: ( , )- ��). Identifiable Problems with the Domain include:A. Dividing by an expression that could, for certain values, be zero:Example 1 Find the domain forx 3f (x)2x 14+=-Can 2x  14 be zero? Throw out the values that do this!2x  14 = 0 2x = 14x = 7Take 7 and say: The domain of this function is all real numbers except 7.In interval notation:( ,7) (7, )-�ȥ.This all real numbers up to and NOT including 7. Remember that parenthesis mean “not including” and square brackets mean “including”.Example 2 Find the domain for 21f (x)x 1=+.2Can the denominator be zero? Throw out the values that do this!22x 1 0x 1x i+ ==-=�No, not in the real numbers, it is greater than zero for any real number.The domain for this function is all real numbers: ( , )- ��.B. Taking the square root* of a math expression that could, for certain values, be a negative number:*or any even-powered rootExample 3 Find the domain forf (x) 15 3x= +.Can 15  x be a negative number?15 + 3 x < 015 < 3xcomputational note: divide an inequality by a negative! 5 > x i.e. x <  5These x’s give negative numbers in the expression. TTO!The domain is [ 5, )- �. (i.e. all numbers greater than or equal to 5).Example 4 Find the domain for f (x) x=.3For which values is the absolute value of x negative?So the domain is all real numbers:( , )- ��.C. Taking a logarithm of an expression that could, for certain values, be zero or a negative number:Example 5 Find the domain for 6f (x) log (x 8)= -.Can x  8 be zero or a negative? Throw out the values that do this! (TTO)x  8 � 0x � 8 these x’s give 0 or a negative number for x  8…The domain is all real numbers strictly greater than 8.In interval notation:(8, )�.Example 6 Find the domain for f (x) log x .=Can the absolute value of x be zero or a negative? It can be zero when x = 0. TIO!The domain is all real numbers except zero.In interval notation:( ,0) (0, )-�ȥ.So how does this really work? You look at the given function and ask yourself the following 3 questions. If the answers are “no, no, no”. The domain is all real numbers.If there’s a yes, stop there and find the domain by the technique of finding the problem x’s and throwing them out (TTO!).The Three Questions:1. Is there denominator that could be equal to zero?2. Is there an even root?3. Is there a logarithm?Example 7 Find the domain for2 x5f (x) x 6 x= + -.4Question 1 denominator?Question 2 even root?Question 3 logarithm?So what is the domain?Example 8 Find the domain for 9f (x)(x 2)(x 1)=+ -.Question 1 denominator?So what is the domain?Another key skill with functions is evaluation. You will be expected to be able to evaluate functions readily.Example 9 Evaluatexf (x) 2=evaluate f(1).Take the x and substitute it in the expression and simplify.Example 10 Evaluate2f (x) x x= - evaluate f(1)Example 11 Evaluatef (x) 2x 7= +evaluate f(x  3)Example 12 Evaluate6f (x) 2x= +evaluate f(x + h)5Example 13 Evaluate f(3), f(0), f(4), and f(6) for2x2x 1 x 5f (x) x 3 x 53 x 3- ���= - - < <���-�Piece-wise defined functions are used a lot in Calculus 1. Get to know this format!What is another name for f(0)?The Difference Quotient:In Calculus you will use the Difference Quotient to find the derivative for a while. It’s important to know how to calculate the Difference Quotient so that you can, when you get to Calculus, calculate the limit of it!Difference means subtract and quotients come up in division. Look at the formula for theDifference Quotient and see these parts:f (x h) f (x)DQh+ -=(you do not need to memorize the formula!)Now, the steps are quite simple but not always easy!Step 1 calculate f(x + h)Step 2 subtract f(x) from the calculation in Step 1Step 3 divide the result by h.Example 14 Calculate the DQ for f(x) = 3x  96Example 15 Calculate the DQ for f(x) = 2x 2x 5- +7Example 16 Evaluate the DQ from Example 15 for x = 2.Example 17 Evaluate the Difference Quotient for 2f (x) 15x= + for x = 5.8Functions and GraphsThe graph of a function is an illustration of the solution set using ordered pairs of coordinates where the first coordinate is from the domain and the second coordinate is theassociated range value.You may use the Vertical Line Test (VLT) to determine if a given graph is that of a function. A vertical line should touch the graph in at most one point. (it’s ok if there’s NO point of intersection, one point of intersection, but NOT 2 or more points of intersection).Here’s some examples – tell me if the graph is drawn from a function and give the domain and range if it is a function, else say “none”.Example 18 A - EA.Is this a function?What is the formula for this?What are the domain and range?9B. Is this a function?What is the formula for this?What are the domain and range?C. Is this a function?Compare to the definition of function…10What is this? Is it a function?What are the parts of this?Domain, range, intercepts…E. Let’s look at this one more closely:List the domain, the range, and the intercepts.f(0)f(3)f(5)11Vocabulary Review:“Value” refers to RANGE values  maximum value What is the maximum value in Example 18 E?  minimum value What is