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Domain, Range, and Function Properties for a Given Relation

This concept explains how to determine the domain, range, and function properties of a given relation. It also demonstrates the use of the vertical line test and graphing a relation. Furthermore, it provides examples of evaluating and identifying linear functions.

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Domain, Range, and Function Properties for a Given Relation

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  1. Splash Screen

  2. Concept 1

  3. Domain and Range State the domain and range of the relation. Then determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. The relation is {(1, 2), (3, 3), (0, –2), (–4, –2), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 1, 2, 3}. Each member of the domain is paired with one member of the range, so this relation is a function. It is onto, but not one-to-one. Example 1

  4. Concept 2

  5. Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points. Example 2

  6. Graph a Relation Find the domain and range. Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that every real number is the x-coordinate of some point on the line. Also, every real number is the y-coordinate of some point on the line. Answer: The domain and range are both allreal numbers. Example 3

  7. Evaluate a Function A. Given f(x) = x3 – 3, find f(–2). f(x) = x3 – 3 Original function f(–2) = (–2)3 – 3 Substitute. = –8 – 3 or –11 Simplify. Answer:f(–2) = –11 Example 4A

  8. Evaluate a Function B. Given f(x) = x3 – 3, find f(2t). f(x) = x3 – 3 Original function f(2t) = (2t)3 – 3 Substitute. = 8t3 – 3 (2t)3 = 8t3 Answer:f(2t) = 8t3 – 3 Example 4B

  9. A linear function is a function with ordered pairs that satisfy a linear function. Any linear function can be written in the form f(x) = mx + b.

  10. Identify Linear Functions B.State whether p(x) = x3 + 2 is a linear function. Write yes or no. Explain. Answer: No; this is not a linear function because x has an exponent other than 1. Example 1B

  11. Identify Linear Functions C.State whether t(x) = 4 + 7x is a linear function. Write yes or no. Explain. Answer: Yes; this is a linear function because it can be written as t(x) = mx + b; m = 7, b = 4. Example 1C

  12. Evaluate a Linear Function A. METEOROLOGYThe linear function f(C) = 1.8C + 32 can be usedto find the number of degrees Fahrenheit f(C) that are equivalent to a given number of degrees Celsius C. On the Celsius scale,normal body temperature is 37C.What is it in degrees Fahrenheit? f(C) = 1.8C + 32 Original function f(37) = 1.8(37) + 32 Substitute. = 98.6 Simplify. Answer: Normal body temperature, in degrees Fahrenheit, is 98.6°F. Example 2

  13. Concept

  14. Use Intercepts to Graph a Line Find the x-intercept and the y-intercept of the graph of –2x + y – 4 = 0. Then graph the equation. The x-intercept is the value of x when y = 0. –2x + y– 4 = 0 Original equation –2x + 0– 4 = 0 Substitute 0 for y. –2x = 4 Add 4 to each side. x = –2 Divide each side by –2. The x-intercept is –2. The graph crosses the x-axis at (–2, 0). Example 4

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