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Section P.3 – Functions and their Graphs

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Functions Algebraic Function Can be written as finite sums, differences, multiples, quotients, and radicals involving x n. Examples: Transcendental Function A function that is not Algebraic. Examples: A relation such that there is no more than one output for each input ABCABC WZWZ

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4 Examples of Functions XY XY These are all functions because every x value has only one possible y value Every one of these functions is a relation.

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3 Examples of Non-Functions XY Every one of these non-functions is a relation. Not a function since x=1 can be either y=10 or y=-3 Not a function since x=-4 can be either y=7 or y=1 Not a function since multiple x values have multiple y values

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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.

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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.

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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. Does not stand for “f times x” It does stand for “plug a value for x into a formula f”

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Example If g(x) = 2x + 3, find g(5). You want x=5 since g(x) was changed to g(5) When evaluating, do not write g(x)! You wanted to find g(5). So the complete final answer includes g(5) not g(x)

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Solving v Evaluating Substitute and Evaluate The input (or x) is 3. Solve for x The output is -5. No equal signEqual sign

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Number Sets Counting numbers ( maybe 0, 1, 2, 3, 4, and so on) Natural Numbers: Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on) Integers: a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on) Rational Numbers: a number that can NOT be expressed as an integer fraction (π, √2, and so on) Irrational Numbers: NONE

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Number Sets The set of all rational and irrational numbers Real Numbers: Natural Numbers Integers Rational Numbers Irrational Numbers Real Number Venn Diagram:

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Set Notation Not Included The interval does NOT include the endpoint(s) Interval NotationInequality NotationGraph Parentheses ( ) < Less than > Greater than Open Dot Included The interval does include the endpoint(s) Interval NotationInequality NotationGraph Square Bracket [ ] ≤ Less than ≥ Greater than Closed Dot

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Graphically and algebraically represent the following: All real numbers greater than 11 Graph: Inequality: Interval: Example Infinity never ends. Thus we always use parentheses to indicate there is no endpoint.

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Describe, graphically, and algebraically represent the following: Description: Graph: Interval: Example All real numbers greater than or equal to 1 and less than 5

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Describe and algebraically represent the following: Describe: Inequality: Symbolic: Example All real numbers less than -2 or greater than 4 The union or combination of the two sets.

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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. The domain and range help determine the window of a graph. x y f

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Example 1 Domain: Range: Domain: Range: Describe the domain and range of both functions in interval notation:

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Example 2 Find the domain and range of. t h ER DOMAIN:RANGE: The range is clear from the graph and table. The input to a square root function must be greater than or equal to 0 Dividing by a negative switches the sign The domain is not obvious with the graph or table.

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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:

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Example 1 Write a piecewise function for each given graph.

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Example 2 Rewrite as a piecewise function. Find the x value of the vertex Change the absolute values to parentheses. Plus make the one on the left negative. x f(x)f(x) Use a graph or table to help.

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Basic Types of Transformations ( h, k ): The Key Point When negative, the original graph is flipped about the x-axis When negative, the original graph is flipped about the y-axis Horizontal shift of h units Vertical shift of k units Parent/Original Function: A vertical stretch if |a|>1and a vertical compression if |a|<1

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Transformation Example Shift the parent graph four units to the left and three units down. Description: Use the graph of below to describe and sketch the graph of.

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Composition of Functions f g First Second OR Substituting a function or it’s value into another function. There are two notations: (inside parentheses always first)

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Example 1 Let and. Find: Substitute x=1 into g(x) first Substitute the result into f(x) last

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Example 2 Let and. Find: Substitute x into f(x) first Substitute the result into g(x) last

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Even v Odd Function Symmetrical with respect to the y-axis. Even Function 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. Symmetrical with respect to the origin. Odd Function

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Example Is the function odd, even, or neither? Check out the graph first. Test by Replacing x in the function by –x. The equation is even. An equivalent equation.

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Delta x Δ x stands for “the change in x.” It is a variable that represents ONE unknown value. For example, if x 1 = 5 and x 2 = 7 then Δ x = 7 – 5 = 2. Δ x can be algebraically manipulated similarly to single letter variables. Simplify the following statements:

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