Lesson 1.2, pg. 138 Functions & Graphs Objectives: To identify relations and functions, evaluate functions, find the domain and range of functions, determine whether a graph is a function, and graph a function.

Domain & Range A relation is a set of ordered pairs. Domain: first components in the relation (independent); x-values Range: second components in the relation (dependent, the value depends on what the domain value is); y-values

Find the domain and range of the relation. {(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}

FUNCTIONS Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element. Every “x” has only one “y”.

Is this relation a function? Mapping – illustrates how each member of the domain is paired with each member of the range (Note: List domain and range values once each, in order.) Draw a mapping for the following. (5, 1), (7, 2), (4, -9), (0, 2) x y Is this relation a function? 4 5 7 -9 1 2

See Example 2, page 150. Determine whether each relation is a function: {(1,2), (3,4), (5,6), (5,8)} {(1,2), (3,4), (6,5), (8,5)}

Functions as Equations Determine whether the equation defines y as a function of x. a) b) Solve for y in terms of x. If two or more values of y can be obtained for a given x, the equation is not a function.

Determine if the equation defines y as a function of x. 2x + y = 6 x2 + y2 = 1 x2 + 2y = 10

Evaluating a Function Common notation: f(x) = function Evaluate the function at various values of x, represented as: f(a), f(b), etc. Example: f(x) = 3x – 7 f(2) = f(3 – x) =

If f(x) = x2 – 2x + 7, evaluate each of the following. a) f(-5) b) f(x + 4) c) f(-x) See Example 4, page 143 for additional practice.

Determine if a relation is a function from the graph? Remember: to be a function, an x-value is assigned ONLY one y-value . On a graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line. THUS, the vertical line test! If a vertical line drawn on any part of your graph touches more than one point, it is NOT the graph of a function.

Graphs of Functions Step 1: Graph the relation. (Use graphing calculator or pencil and paper.) Step 2: Use the vertical line test to see if the relation is a function. Vertical line test – If any vertical line passes through more than one point of the graph, the relation is not a function.

Determine if the graph is a function. a) b) y x 5 -5

Here’s more practice. y x c) d) y x

Example Analyze the graph.

Find f(7). (a) (b) (c) (d)

Can you identify domain & range from the graph? Look horizontally. What x-values are contained in the graph? That’s your domain! Look vertically. What y-values are contained in the graph? That’s your range! Write domain and range using interval or set-builder notation. See Example 8, page 148.

Domain: set of all values of x Range: set of all values of y Always write the domain and range in interval notation when reading the domain and range from a graph. Use brackets [ or ] to show values that are included in the graph. Use parentheses ( or ) to show values that are NOT included in the graph.

Identify the Domain and Range from the graph. Example Identify the Domain and Range from the graph.

Identify the Domain and Range from the graph. Example Identify the Domain and Range from the graph.

Identify the Domain and Range from the graph. Example Identify the Domain and Range from the graph.

Find the Domain and Range. (b) (c) (d)

What is the difference in the two sets below, and when should we use each to describe the domain of a function? [1,4] {1,2,3,4}

Finding intercepts: x-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero. y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero. Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!

Find the x intercept(s). Find f(-4) Example Find the x intercept(s). Find f(-4)

Find the x and y intercepts. Find f(5). Example Find the x and y intercepts. Find f(5).

Summary Domain = x values Range = y values Use the vertical line test to verify if a graph is a function. To evaluate means to substitute and simplify. Intercepts – where function crosses the x-or y-axis