2 FunctionsA relation such that there is no more than one output for each inputABCWZAlgebraic FunctionCan be written as finite sums, differences, multiples, quotients, and radicals involving xn.Examples:Transcendental FunctionA function that is not Algebraic.
3 Every one of these functions is a relation. 4 Examples of FunctionsXY10215-5182017These are all functions because every x value has only one possible y valueXY-31-14573Every one of these functions is a relation.
4 3 Examples of Non-Functions Not a function since x=-4 can be either y=7 or y=1XY4110211-353Not a function since multiple x values have multiple y valuesNot a function since x=1 can be either y=10 or y=-3Every one of these non-functions is a relation.
5 The Vertical Line TestIf a vertical line intersects a curve more than once, it is not a function.Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.
6 The Vertical Line TestIf a vertical line intersects a curve more than once, it is not a function.Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.
7 Function Notation: f(x) Equations that are functions are typically written in a different form than “y =.” Below is an example of function notation: The equation above is read: f of x equals the square root of x. The first letter, in this case f, is the name of the function machine and the value inside the parentheses is the input. The expression to the right of the equal sign shows what the machine does to the input.It does stand for “plug a value for x into a formula f”Does not stand for “f times x”
8 Example If g(x) = 2x + 3, find g(5). When evaluating, do not write g(x)!You want x=5 since g(x) was changed to g(5)You wanted to find g(5). So the complete final answer includes g(5) not g(x)
9 Substitute and Evaluate Solving v EvaluatingNo equal signEqual signSubstitute and EvaluateThe input (or x) is 3.Solve for xThe output is -5.
10 Number Sets Natural Numbers: Counting numbers (maybe 0, 1, 2, 3, 4, and so on)Integers:Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on)Rational Numbers:a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on)a number that can NOT be expressed as an integer fraction (π, √2, and so on)Irrational Numbers:NONE
11 Number Sets Real Numbers: The set of all rational and irrational numbersRational NumbersIntegersIrrationalNumbersReal Number Venn Diagram:Natural Numbers
12 Set Notation Not Included The interval does NOT include the endpoint(s)Interval NotationInequality NotationGraphParentheses( )< Less than> Greater thanOpen DotIncludedThe interval does include the endpoint(s)Interval NotationInequality NotationGraphSquare Bracket[ ]≤ Less than≥ Greater thanClosed Dot
13 All real numbers greater than 11 Example 1Graphically and algebraically represent the following:All real numbers greater than 11Graph:Inequality:Interval:Infinity never ends. Thus we always use parentheses to indicate there is no endpoint.
14 Example 2 1 3 5 Description: Graph: Interval: Describe, graphically, and algebraically representthe following:Description:Graph:Interval:All real numbers greater than or equal to 1 and less than 5
15 The union or combination of the two sets. Example 3Describe and algebraically represent thefollowing:Describe:Inequality:Symbolic:All real numbers less than -2 or greater than 4The union or combination of the two sets.
16 The domain and range help determine the window of a graph. All possible input values (usually x), which allows the function to work.RangeAll possible output values (usually y), which result from using the function.fxyThe domain and range help determine the window of a graph.
17 Example 1Describe the domain and range of both functions in interval notation:Domain:Domain:Range:Range:
18 Example 2 Find the domain and range of . The domain is not obvious with the graph or table.The input to a square root function must be greater than or equal to 0Dividing by a negative switches the signt-32-20-155-4123h1087-74ERThe range is clear from the graph and table.DOMAIN:RANGE:
19 Piecewise FunctionsFor Piecewise Functions, different formulas are used in different regions of the domain. Ex: An absolute value function can be written as a piecewise function:
20 Example 1Write a piecewise function for each given graph.
21 Example 2 Rewrite as a piecewise function. Use a graph or table to help.x-3-2-11234f(x)65Find the x value of the vertexChange the absolute values to parentheses. Plus make the one on the left negative.
22 Basic Types of Transformations Parent/Original Function:When negative, the original graph is flipped about the x-axisA vertical stretch if |a|>1and a vertical compression if |a|<1Horizontal shift of h unitsWhen negative, the original graph is flipped about the y-axisVertical shift of k units( h, k ): The Key Point
23 Transformation Example Use the graph of below to describe and sketch the graph ofDescription:Shift the parent graph four units to the left and three units down.
24 Composition of Functions Substituting a function or it’s value into another function. There are two notations:gSecondFirstORf(inside parentheses always first)
25 Example 1 Let and . Find: Substitute x=1 into g(x) first Substitute the result into f(x) last
26 Example 2 Let and . Find: Substitute the result into g(x) last Substitute x into f(x) first
27 Even v Odd Function Even Function Odd Function Symmetrical with respect to the y-axis.Symmetrical with respect to the origin.Tests…Replacing x in the function by –x yields an equivalent function.Replacing x in the function by –x yields the opposite of the function.
28 Example The equation is even. Is the function odd, even, or neither? Test by Replacing x in the function by –x.Check out the graph first.An equivalent equation.The equation is even.
29 Delta xΔx stands for “the change in x.” It is a variable that represents ONE unknown value. For example, if x1 = 5 and x2 = 7 then Δx = 7 – 5 = 2. Δx can be algebraically manipulated similarly to single letter variables. Simplify the following statements: