The document discusses representing relations and functions through mapping diagrams, graphs, and equations. It defines relations as mappings of inputs to outputs, with domains as the set of inputs and ranges as the set of outputs. Functions are special relations where each input maps to exactly one output. Linear functions can be written in the form y=mx+b and graphed on a coordinate plane. Both relations and functions have restrictions on their domains in real-world applications to make them reasonable.

1. 2.1 Represent Relations and Functions A relation is a mapping, or pairing, of input values with output values. Domain – the set of input values. Range – the set of output values.

2. 2.1 Represent Relations and Functions

3. 2.1 Represent Relations and Functions Example 1 Consider the relation given by (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3) Identify the domain and range. Represent the relation using a graph and a mapping diagram.

4. 2.1 Represent Relations and Functions A Function is a relation for which each input has exactly one output. If any input has more than one output the relation is not a function. A function is always a relation. A relation is not always a function.

5. 2.1 Represent Relations and Functions Example 2 ANIMATION

6. 2.1 Represent Relations and Functions

7. 2.1 Represent Relations and Functions

8. 2.1 Represent Relations and Functions

9. 2.1 Represent Relations and Functions Many functions can be described as a function with two variables. Example: y = 3x – 5 The input variable is x, also known as independent variable. The output variable is y, also known as dependent variable. The output depends on the input. An ordered pair is a solution of an equation in two vairables. A graph is another way to represent the solutions of the equation.

10. 2.1 Represent Relations and Functions

11. 2.1 Represent Relations and Functions Example 4 Graph the equation y = 3x – 5. What is the minimum number of points you need? Where does the graph cross the y-axis?

12. 2.1 Represent Relations and Functions A function that can be written in the form y = mx + b where m and b are constants is called a linear function. When y is replaced by f(x) the function is written using function notation. y = mx + b  Linear function in x-y notation f(x) = mx + b Linear function in function notation. The notation f(x) is read “the value of f at x” or “f of x” and identifies x as the independent variable.

13. 2.1 Represent Relations and Functions Example 5 Tell whether the function is linear. Then evaluate the function when x = -3. What is the domain & range? f(x) = -2x3 + 5 g(x) = 12 – 8x

14. 2.1 Represent Relations and Functions Graph y = 3x - 2

15. 2.1 Represent Relations and Functions

16. 2.1 Represent Relations and Functions In real life, you may need to restrict domain so that it is reasonable to the given situation. To do this inequalities are used.

17. 2.1 Represent Relations and Functions Example 6 The length L (in inches) that a spring stretches when a weight up to 20 pounds is attached to it is given by L(w) = 1/12w + 2, where w is the weight in pounds. Graph the function and determine a reasonable domain and range. What is the length of the spring when a 10 pound weight is attached?

18. 2.1 Represent Relations and Functions In 1960, the deep sea vessel Trieste descended to an estimated depth of 35,800 feet. Determine a reasonable domain and range of the function P(d) for this trip. P(d) = 1 + 0.03d

19. 2.1 Represent Relations and Functions How do you graph relations and functions?

20. 2.1 Extension

21. 2.1 Extension Example 1 Graph y = -x + 3 for x ≥ -1. Classify the function as discrete or continuous. Identify the range.

22. 2.1 Extension Example 2 Mrs. Malone buys paint for $20 per gallon. The function f(x) gives the cost of buying x gallons of paint. Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

23. 2.1 Extension How do the graphs of discrete functions and continuous functions differ?