Functions Topic 8.5.2

Functions 8.5.2 1.1.1 California Standards: What it means for you: Lesson 1.1.1 Topic 8.5.2 Functions California Standards: 16.0: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 18.0: Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. What it means for you: You’ll find out what functions are, and you’ll say whether particular relations are functions. Key words: function relation ordered pair domain range

Lesson 1.1.1 Topic 8.5.2 Functions In Topic 8.5.2 you learnt about relations — a relation is a set of ordered pairs. In this Topic, you’ll learn about functions — a function is a special type of relation.

Functions 8.5.2 1.1.1 Functions Map from the Domain to the Range Lesson 1.1.1 Topic 8.5.2 Functions Functions Map from the Domain to the Range A relation is any set of ordered pairs — without restriction. A function is a type of relation that has the following restriction on it: A function is a set of ordered pairs (x, y) such that no two ordered pairs in the set have the same x-value but different y‑values. That is, each member of the domain maps to a unique member of the range.

Topic 8.5.2 Functions Example 1 Determine whether each of the following relations is a function or not. Justify your answers. a) k = {(0, 0), (1, 1), (2, 4), (3, 9)} b) m = {(1, 2), (2, 5), (1, 4), (3, 6)} c) p = {(–1, –3), (0, –1), (1, 1), (2, 3)} d) v = {(–2, 5), (–1, 5), (0, 5), (1, 5), (5, 5)} Solution a) k is a function. No two different ordered pairs have the same x-value. b) m is not a function. The ordered pairs (1, 2) and (1, 4) have the same first entry, but different second entries. Solution continues… Solution follows…

Topic 8.5.2 Functions Example 1 Determine whether each of the following relations is a function or not. Justify your answers. a) k = {(0, 0), (1, 1), (2, 4), (3, 9)} b) m = {(1, 2), (2, 5), (1, 4), (3, 6)} c) p = {(–1, –3), (0, –1), (1, 1), (2, 3)} d) v = {(–2, 5), (–1, 5), (0, 5), (1, 5), (5, 5)} Solution (continued) c) p is a function. No two different ordered pairs have the same x-value. d) v is a function. No two different ordered pairs have the same x-value.

Functions 8.5.2 1.1.1 Guided Practice Lesson 1.1.1 Topic 8.5.2 Functions Guided Practice State whether each relation in Exercises 1–4 is a function or not. Explain your reasoning. 1. m = {(1, 1), (2, 8), (3, 27)} 2. b = {(a, 1), (b, 2), (c, 3), (a, 4)} 3. v = {(7, 1), (7, 2), (7, 7)} 4. t = {(1, 7), (2, 7), (7, 7)} m is a function because each first entry is unique b is not a function since (a, 1) and (a, 4) have the same first entry, but different second entries v is not a function because each first entry is identical (and the second entries are different) t is a function because each first entry is unique Solution follows…

8.5.2 1.1.1 Functions Functions Can Be Represented in Different Ways Topic 8.5.2 Lesson 1.1.1 Functions Functions Can Be Represented in Different Ways Relations do not have to be written as lists of ordered pairs for you to be able to identify functions. You can identify functions from mapping diagrams.

Topic 8.5.2 Functions Example 2 Determine whether each of the mappings below represents a function. a) b) c) a b c k n 2 5 7 11 a b c k n 2 5 6 7 11 a b c n 2 5 6 7 11 Solution a) and b) are NOT functions, since 7 is mapped to two different values — (7, a) and (7, c) have the same x-value. c) IS a function. Each member of the domain only maps onto one member of the range. Solution follows…

Functions 8.5.2 1.1.1 Guided Practice Lesson 1.1.1 Topic 8.5.2 Functions Guided Practice The mapping below shows the relation g(x). 5. State the domain and range of the relation. 6. Is the relation a function? Explain your answer. 7. Find g(0). Domain = {–4, –1, 0, 1, 2} Range = {–1, 2, 3, 4, 5} g(x) is a function, as each element in the domain maps onto one member of the range. g(0) = 3 Solution follows…

Functions 8.5.2 1.1.1 Guided Practice Lesson 1.1.1 Topic 8.5.2 Functions Guided Practice The mapping below shows the relation h(x). 8. State the domain and range of the relation. 9. Is this relation a function? Explain your answer. 10. Find h(–3). Domain = {–4, –3, 2, 5} Range = { p, q, r, s, t } h(x) is not a function, since –4 and 5 in the domain each map to two different members of the range. h(–3) = s Solution follows…

Functions 8.5.2 1.1.1 Functions are Often Written as Equations Lesson 1.1.1 Topic 8.5.2 Functions Functions are Often Written as Equations Some functions can be expressed as an equation. For this to be possible, there must be a reason for the pairing between each member of the domain and each member of the range. The equation represents the way the members of the domain and range are paired.

Topic 8.5.2 Functions Example 3 Express the following function as an equation: f = {(1, 1), (2, 4), (3, 9), (4, 16)} Solution The relationship between the x-values and y-values is y = x2. The domain of the function is x  {1, 2, 3, 4}. The function can be written: f = {(x, y) such that y = x2 and x  {1, 2, 3, 4}} Solution follows…

Lesson 1.1.1 Topic 8.5.2 Functions You will often see functions written in the form y = x2, without any domain specified. By convention, you then take the domain to be all values of x for which the function is defined.

Functions 8.5.2 1.1.1 Guided Practice Lesson 1.1.1 Topic 8.5.2 Functions Guided Practice Express the following functions in terms of an equation. 11. f = {(–4, 0), (–3, 1), (0, 4), (1, 5), (2, 6)} 12. g = {(–2, 5), (0, 1), (1, 2), (2, 5)} 13. h = {(–5, –4.5), (–3, –2.5), (1, 1.5), (3, 3.5), (5, 5.5)} f = {(x, y) such that y = x + 4 and x  {–4, –3, 0, 1, 2}} g = {(x, y) such that y = x2 + 1 and x  {–2, 0, 1, 2}} h = {(x, y) such that y = x + ½ and x  {–5, –3, 1, 3, 5}} Solution follows…

Functions 8.5.2 1.1.1 Guided Practice Lesson 1.1.1 Topic 8.5.2 Functions Guided Practice Express the following functions in terms of an equation. 14. f = {(–3, –27), (–2, –8), (–1, –1), (0, 0), (1, 1), (2, 8)} 15. g = {(–2, 8), (0, 0), (1, 2), (2, 8), (3, 18)} 16. h = {(–3, 17), (–1, 1), (0, –1), (1, 1), (2, 7)} f = {(x, y) such that y = x3 and x  {–3, –2, –1, 0, 1, 2}} g = {(x, y) such that y = 2x2 and x  {–2, 0, 1, 2, 3}} h = {(x, y) such that y = 2x2 – 1 and x  {–3, –1, 0, 1, 2}} Solution follows…

Topic 8.5.2 Lesson 1.1.1 Functions The Vertical Line Test Shows if a Graph is a Function By definition, a function cannot have any two ordered pairs that have the same first coordinate but different second coordinates, i.e. for each value of x there is only one possible value of y. Graphically, this means that no vertical line can intersect the graph of a function at more than one point.

Lesson 1.1.1 Topic 8.5.2 Functions Vertical Line Test to determine whether a graph represents a function: Simply hold a straightedge parallel to the y-axis at the far left-hand side of the graph, then move it horizontally along the graph from left to right. If, at any position along the x-axis, it is possible for you to draw a vertical line that intersects with the graph more than once, then the graph does not represent a function.

Topic 8.5.2 Functions Example 4 Use the vertical line test to determine whether the following graphs represent functions. Solution The first two graphs pass the vertical line test — you cannot draw a vertical line that intersects with either graph at more than one point, so they represent functions. Solution continues… Solution follows…

Topic 8.5.2 Functions Example 4 Use the vertical line test to determine whether the following graphs represent functions. Solution (continued) The third graph does not represent a function — the vertical line test fails. The line intersects the graph more than once.

8.5.2 1.1.1 Functions Guided Practice Topic 8.5.2 Lesson 1.1.1 Functions Guided Practice In Exercises 17–20, use the vertical line test to determine whether each graph represents a function or not. 17. 18. A function Not a function Not a function 19. 20. A function Solution follows…

Functions 8.5.2 Independent Practice Topic 8.5.2 Functions Independent Practice 1. Define a function and give an example. In Exercises 2–7, use the given relation with the domain x = {–2, –1, 0, 1, 2} to generate sets of ordered pairs. Use them to determine whether the relation is a function or not. Sample answer: A function is any relation in which the first entry of each distinct ordered pair is unique. Example: k = {(–3, 9), (–2, 4), (–1, 1), (0, 1), (1, 1), (2, 4)} is an example of a function since the first entry in each ordered pair is unique. 2. m = {(x, x2 – 4)} 3. t = {(x, x + 2)} 4. k = {(x, y = (x – 2)(x + 2)} 5. p = {(x, y = ± } 6. j = {(x, y = 2x – 1)} 7. b = {(x, y = x ± (3x – 4)} A function A function A function Not a function A function Not a function Solution follows…

Functions 8.5.2 Independent Practice Topic 8.5.2 Functions Independent Practice 8. In the equation x2 + y2 = 9, is y a function of x? Explain your reasoning. Using the graph below, answer Exercises 9–12 about relation f. y is not a function of x because , which means that there are ordered pairs of the relation that have the same x-value, but different y-values. 9. State the domain and range of the relation. 10. Is the relation a function? 11. Find the value(s) of f (0). 12. Find the value(s) of f (3). Domain = {–3 ≤ x ≤ 3} Range = {–3 ≤ f (x) ≤ 3} No f (0) = 3 and –3 f (3) = 0 Solution follows…

Functions 8.5.2 Independent Practice 13. 14. Topic 8.5.2 Functions Independent Practice Use the vertical line test to determine if the graphs below are functions. 13. 14. A function Not a function For Exercises 15–16, find the range (y) for each relation when the domain is {–4, –2, 0, 2, 4}, and determine whether the relation is a function. 15. y = x + 1 16. y = x – 6 {–1, 0, 1, 2, 3} A function {–10, –8, –6, –4, –2} A function Solution follows…

No — they don’t pass the vertical line test Topic 8.5.2 Functions Independent Practice For Exercises 17–19, find the domain (x) for each relation when the range is {–6, –3, 0, 3, 6}, and determine whether the relation is a function. 17. y = x – 2 18. y = x + 5 19. y = {–12, –3, 6, 15, 24} A function {–11, –8, –5, –2, 1} A function {–6, –3, 0, 3, 6} Not a function 20. Are all quadratics of the form y = ax2 + bx + c and y = –ax2 + bx + c functions? Explain your answer. Yes — they are either concave up or down with only one y-value for each x-value 21. Are circles functions? Explain your answer. No — they don’t pass the vertical line test Solution follows…

Functions 8.5.2 Round Up Functions are special types of relations. Topic 8.5.2 Functions Round Up Functions are special types of relations. That means that all functions are relations — but not all relations are functions. A relation is only a function if it maps each member of the domain to only one member of the range.