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Understanding Linear Equations: Domain, Range & Functions

Learn about linear equations, relations, functions, domain, range, and function notation in Advanced Algebra & Trigonometry. Analyze graphs, equations, and ordered pairs to determine if a relation is a function. Practice finding domain and range of various relations.

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Understanding Linear Equations: Domain, Range & Functions

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  1. Do Now Complete the chart for each linear equation. y = x - 2 y = 2x + 3

  2. Advanced Algebra TrigonometryAppendix CFunctions Objective: Determine if a relation is a function, and find the domain and range.

  3. Relations Relation:a set of ordered pairs. { ( -3, 2), (-1, 1 ), ( 0, 7), (2, 4), (4, 3)} {( -2, 1), (-1, 2), ( 0, 3), (1, 4), (2, 5)} ( x, y ) Independent Variable Dependent Variable The value of y “depends” on the value of x. Domain: the set of all x-coordinates, independent variable Range: the set of all y-coordinates, dependent variable

  4. Relations Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4}

  5. Functions Function: a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. • A function is a relation in which the members of the domain (x-values) DO NOT repeat. • So, for every x-value there is only one y-value that corresponds to it. • y-values CAN be repeated.

  6. X Y 1 2 5 10 -1 -2 3 6 Ways to Represent a Function • Graphical • Symbolic • Numeric • VerbalThe cost is twice the original amount.

  7. Does the relation represent a function? H = {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. G = {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated.

  8. Finding Domain & Range Give the domain & range of each relation. Is it a function? Example 1 Example 2 {(3, -6), (1, 3), (-2, 4), (0,3), (1, -2), (3, 0)}

  9. Vertical Line Test Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

  10. Finding Domain & Range Give the domain & range of each relation. Is it a function?

  11. x y Finding Domain & Range Give the domain & range of each relation. Is it a function? . . . .

  12. Function: Yes D: All real numbers R: All real numbers Function: Yes D: All real numbers R: y ≥ -6 x x y y Does the graph represent a function? Name the domain and range.

  13. Function: No D: x ≥ 1/2 R: All real numbers Function: No D: All real numbers R: All real numbers x x y y Does the graph represent a function? Name the domain and range.

  14. Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used. • The parenthesis DO NOT mean multiplication! • f(x) is another name for y. • Sometimes other letters such as g, h or capital letters F, G and H are used to name functions.

  15. Using Function Notation Find the value of each function. • If g(s) = 2s + 3, find g(-2). • If h(x) = x2 - x + 7, find h(2). • If f(x) = -x2 + 5x – 3 find f(q)

  16. Homework Page 457-458 #’s 12, 13, 18, 19, 35, 36, 37, 46

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