Section P.3 – Functions and their Graphs

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

Every one of these functions is a relation. 4 Examples of Functions X Y 10 2 15 -5 18 20 1 7 These are all functions because every x value has only one possible y value X Y -3 1 -1 4 5 7 3 Every one of these functions is a relation.

3 Examples of Non-Functions Not a function since x=-4 can be either y=7 or y=1 X Y 4 1 10 2 11 -3 5 3 Not a function since multiple x values have multiple y values Not a function since x=1 can be either y=10 or y=-3 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: 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”

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)

Substitute and Evaluate Solving v Evaluating No equal sign Equal sign Substitute and Evaluate The input (or x) is 3. Solve for x The output is -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) a number that can NOT be expressed as an integer fraction (π, √2, and so on) Irrational Numbers: NONE

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

All real numbers greater than 11 Example 1 Graphically and algebraically represent the following: All real numbers greater than 11 Graph: Inequality: Interval: 10 11 12 Infinity never ends. Thus we always use parentheses to indicate there is no endpoint.

Example 2 1 3 5 Description: Graph: Interval: Describe, graphically, and algebraically represent the following: Description: Graph: Interval: All real numbers greater than or equal to 1 and less than 5 1 3 5

The union or combination of the two sets. Example 3 Describe and algebraically represent the following: Describe: Inequality: Symbolic: -2 1 4 All real numbers less than -2 or greater than 4 The union or combination of the two sets.

The domain and range help determine the window of a graph. 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: Domain: Domain: Range: Range:

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 0 Dividing by a negative switches the sign t -32 -20 -15 5 -4 1 2 3 h 10 8 7 -7 4 ER The range is clear from the graph and table. DOMAIN: RANGE:

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:

Example 1 Write a piecewise function for each given graph.

Example 2 Rewrite as a piecewise function. Use a graph or table to help. x -3 -2 -1 1 2 3 4 f(x) 6 5 Find the x value of the vertex Change the absolute values to parentheses. Plus make the one on the left negative.

Basic Types of Transformations Parent/Original Function: 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 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 below to describe and sketch the graph of . 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: g Second First OR f (inside parentheses always first)

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

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

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.

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.

Delta x Δx 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: