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Section P.3 – Functions and their Graphs Functions A relation such that there is no more than one output for each input A W B Z C Algebraic Function Can be written as finite sums, differences, multiples, quotients, and radicals involving xn. f x3 x x 1 0 2 Examples: Transcendental Function 1 gx2x 4 4x A function that is not Algebraic. hx sinx Examples: gx lnx 4 Examples of Functions X Y 10 2 15 -5 X Y -3 1 -1 0 0 4 5 7 7 3 18 -5 20 1 7 These are all functions because every x value has only one possible y value -5 Every one of these functions is a relation. 3 Examples of Non-Functions X Y 0 4 1 10 Not a function since x=-4 can be either y=7 or y=1 2 11 1 -3 5 3 Not a function since x=1 can be either y=10 or y=-3 Not a function since multiple x values have multiple y values Every one of these non-functions is a relation. The Vertical Line Test If 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. The Vertical Line Test If 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. Function Notation: f(x) Equations that are functions are typically written in a different form than “y =.” Below is an example of function notation: It does stand for Does not stand f x x “plug a value for x for “f times x” into a formula f” 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. Example If g(x) = 2x + 3, find g(5). g 5 2 5 3 When evaluating, do not write g(x)! You wanted to find g(5). So the complete final answer includes g(5) not g(x) g 5 10 3 g 5 13 You want x=5 since g(x) was changed to g(5) Solving v Evaluating If f x 2 3 x 3, com plete the follow ing: No equal sign Equal sign a. E valuate f 3 x b . F in d x if f Substitute and Evaluate Solve for x The input (or x) is 3. The output is -5. 2 3 3 3 2 3 1 5 2 3 x3 8 2 3 x 12 x 5 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) Irrational Numbers: a number that can NOT be expressed as an integer fraction (π, √2, and so on) NONE Number Sets Real Numbers: The set of all rational and irrational numbers Real Number Venn Diagram: Rational Numbers Integers Irrational Natural Numbers Numbers Set Notation Not Included The interval does NOT include the endpoint(s) Interval Notation Inequality Notation Graph Parentheses ( ) < Less than > Greater than Open Dot Included The interval does include the endpoint(s) Interval Notation Inequality Notation Graph Square Bracket [ ] ≤ Less than ≥ Greater than Closed Dot Example 1 Graphically and algebraically represent the following: All real numbers greater than 11 Graph: 10 Inequality: Interval: x1 1 11, 11 12 Infinity never ends. Thus we always use parentheses to indicate there is no endpoint. Example 2 Describe, graphically, and algebraically represent the following: 1 x5 Description: All real numbers greater than or equal to 1 and less than 5 Graph: Interval: 1 1,5 3 5 Example 3 Describe and algebraically represent the following: -2 1 4 All real numbers less than -2 or Describe: greater than 4 Inequality: Symbolic: x 2 o r x 4 The union or combination of the two sets. ,2 4 , Domain and Range Domain All possible input values (usually x), which allows the function to work. Range All possible output values (usually y), which result from using the function. f x y The domain and range help determine the window of a graph. Example 1 Describe the domain and range of both functions in interval notation: y x1 x 9 , Domain: 8 ,2 2 ,9 Domain: 2 5 , Range: Range: 7,8 Example 2 Find the domain and range of h t 4 3 t. The domain is not obvious with the graph or table. The input to a square root function must be greater than or equal to 0 4 3 t 0 3 t 4 4 Dividing by a t3 negative switches the sign t -32 -20 -15 5 -4 0 h 10 -7 4 2 8 7 DOMAIN: 1 1 2 3 ER ER RANGE: , 4 3 The range is clear from the graph 0, and table. Piecewise Functions For Piecewise Functions, different formulas are used in different regions of the domain. Ex: An absolute value function can be written as a piecewise function: x ifx0 x x ifx0 Example 1 Write a piecewise function for each given graph. gx f x x 5 if ifx 0 1 x 4 gx 1 f x 0 x 1ifx 7 if x 4 2 Example 2 Rewrite fx as a piecewise function. x 2 1 Use a graph or table to help. x -3 f(x) 6 -2 -1 0 1 2 3 4 5 4 3 2 1 2 3 Find the x value of the vertex Change the absolute values to parentheses. Plus make the one on the left negative. 2 1ifx x 2 f x x 2 1 ifx 2 Basic Types of Transformations Parent/Original Function: yf x When negative, the original graph is flipped about the x-axis A vertical stretch if |a|>1and a vertical compression if |a|<1 Horizontal shift of h units y a f x h k When negative, the original graph is flipped about the y-axis Vertical shift of k units ( h, k ): The Key Point Transformation Example Use the graph of y x below to describe and 1 sketch the graph of y . 3 x 4 1 Description: Shift the parent graph four units to the left and three units down. Composition of Functions Substituting a function or it’s value into another function. There are two notations: f gx Second OR First f gx (inside parentheses always first) g f Example 1 2 g x x 5 Let f and . Find: x 2 x 3 f g1 g 1 1 5 2 Substitute x=1 into g(x) first f 4 2 4 3 1 5 8 3 4 4 11 f g 1 1 1 Substitute the result into f(x) last Example 2 2 g x x 5 Let f and . Find: x 2 x 3 g f x x Substitute the result into g(x) last g 2 x 3 2 x 3 5 2 Substitute x into f(x) first f x 2 x 3 2 x 3 2 x 3 2 x 3 5 4 x 1 2 x 9 5 2 4 x 1 2 x 9 5 2 4 x1 2 x 4 2 g f x 4 x 1 2 x 4 2 Even v Odd Function Even Function Tests… Symmetrical with respect to the y-axis. f x f x Replacing x in the function by –x yields an equivalent function. Odd Function Symmetrical with respect to the origin. f x f x Replacing x in the function by –x yields the opposite of the function. Example Is the function f x x odd, even, or neither? Check out the graph first. 4 Test by Replacing x in the function by –x. f x x f x x 4 4 An equivalent equation. The equation is even. Delta x x 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: Δ 1 . 5x 1 2 5 x 1 5 x 1 2 5 x 5 x 5 x 1 2 2 5 1 0 x 1 x 2 . x xx x x xx x x x 2 1