We can evaluate algebraic expressions when given the values of the variables.

Transcription

1 5.2 Skill Builder Evaluating Expressions We can evaluate algebraic expressions when given the values of the variables. To evaluate 7k 2 for k 4: 7k 2 7(4) k means 7 k. 1. Evaluate each expression when x 3. a) 3x 4 b) 5x 7 3( ) 4 4 Solving Equations To solve an equation, find the value of the variable that makes the equation true. To solve 3x 4 28: 3x 4 28 Isolate 3x: subtract 4 from each side. 3x x 24 Divide each side by 3. x 8 1. Solve each equation. a) 3x x 4 23 x b) 5x 7 42 x Copyright 2011 Pearson Canada Inc. 257

2 5.2 Properties of Functions FOCUS Develop an understanding of functions. The set of 1st elements of a relation is its domain. The set of related 2nd elements of a relation is its range. A function is a special type of relation where each element in the domain is associated with exactly one element in the range. Example 1 Identifying Functions For each relation below: Identify its domain and range. Decide whether the relation is a function. a) A relation that associates 5 foods to the food groups to which they belong: {(orange, fruit), (cheese, dairy), (broccoli, vegetable), (milk, dairy), (kiwi, fruit)} b) is the number of players on a team for baseball basketball hockey soccer volleyball Solution a) {(orange, fruit), (cheese, dairy), (broccoli, vegetable), (milk, dairy), (kiwi, fruit)} The domain is the set of 1st elements of the ordered pairs: {orange, cheese, broccoli, milk, kiwi} When we list the elements of the domain and range, we do The range is the set of 2nd elements of the ordered pairs: not repeat an element that {fruit, dairy, vegetable} occurs more than once. to see if any ordered pairs have the same 1st element: {(orange, fruit), (cheese, dairy), (broccoli, vegetable), (milk, dairy), (kiwi, fruit)} Each ordered pair has a different 1st element. So, the relation is a function. b) The domain is the set of elements in the 1st set: {5, 6, 9, 11} The range is the set of elements in the 2nd set: {baseball, basketball, hockey, soccer, volleyball} to see if there is more than one arrow from any element in the 1st set. Since there are two arrows from 6 in the 1st set, the relation is not a function Copyright 2011 Pearson Canada Inc.

3 1. For each relation below: Identify its domain and range. Decide whether the relation is a function. a) A relation that associates the numbers of tickets required for different rides at Galaxyland in the West Edmonton Mall: {(4, Cosmo s Space Derby), (6, Galaxy Twister), (7, Mindbender), (4, Galaxyland Raceway), (3, Balloon Race)} The domain is the set of 1st elements: The range is the set of 2nd elements: ordered pairs have the same 1st element: So, the relation a function. b) is in the key of alto saxophone clarinet French horn piano trombone B b C E b F The domain is: The range is: There is arrow from each element in the 1st set. So, the relation a function. This table shows the masses of different numbers of Canadian quarters. independent variable Number of Quarters, n Mass, m (g) dependent variable domain range The mass of the quarters, m, depends on the number of quarters, n. So, we say m is the dependent variable and n is the independent variable Copyright 2011 Pearson Canada Inc. 259

4 Example 2 Describing Functions This table shows sample costs for a pay-as-you-go cell phone plan. Number of Minutes, n Cost, C ($) A table of values usually represents a sample of the ordered pairs in a relation. a) Why is this relation also a function? b) Identify the dependent variable and the independent variable. c) Write the domain and range. Solution a) No two numbers in the 1st column are the same. So, the relation is a function. b) The cost, C, depends on the number of minutes, n. So, C is the dependent variable and n is the independent variable. c) The 1st column of the table represents the domain. The symbol shows The domain is: {10, 20, 30, 40, 50, } that the domain and The 2nd column of the table represents the range. range may continue. The range is: {2, 4, 6, 8, 10, } 1. This table shows the Calories burned for various running times, at an average speed of 8 km/h. Number of Minutes, n Calories Burned per Kilogram, C a) Why is this relation also a function? Copyright 2011 Pearson Canada Inc.

5 b) Identify the dependent variable and the independent variable. The depends on So, is the dependent variable and is the independent variable. c) Write the domain and range. The domain is: The range is: We can write an equation that represents a function using function notation. For example, to show that C 15 2n represents a function, We say: C of n is equal we write: C(n) 15 2n to 15 2n. This notation shows that C is the dependent variable and that C depends on n. Example 3 Using Function Notation to Find Values Carmen works for a research company in a shopping mall. The equation P 5n 30 represents her daily pay, P dollars, when she conducts n surveys. a) Describe the function. Write the equation using function notation. b) Find the value of P(8). c) Find the value of n when P(n) = 90. Solution a) Carmen s pay is a function of the number of surveys she conducts. In function notation: P(n) 5n 30 b) To find P(8), use: P(n) 5n 30 Substitute: n 8 P(8) 5(8) 30 P(8) P(8) is the value of P when n 8. P(8) 70 This means that when Carmen conducts 8 surveys, she earns $70. c) To find the value of n when P(n) 90, use: P(n) 5n 30 Substitute: P(n) = n 30 Solve for n. Subtract 30 from each side n n Divide each side by n 5 5 n 12 P(n) 90 when n 12 This means that when Carmen conducts 12 surveys, she earns $ Copyright 2011 Pearson Canada Inc. 261

6 1. Frank sells memberships to a local gym. The equation E 50n 150 represents his weekly earnings, E dollars, when he sells n memberships. a) Describe the function. Write the equation using function notation. are a function of. In function notation: E 150 b) Find the value of E(9). E(n) 50n 150 Substitute: n 9 E(9) 50( ) 150 E(9) E(9) This means that when Frank sells memberships, he earns. c) Find the value of n when E(n) 850. E(n) 50n 150 Substitute: E(n) n 150 n E(n) 850 when n This means that when Frank sells memberships, he earns. Practice 1. For each relation below, decide whether the relation is a function. How do you know? a) 4 is the number of letters in the word cube So, the relation a function. 6 pentagon rectangle 8 square 9 triangle Copyright 2011 Pearson Canada Inc.

7 b) A relation that associates cities to famous people who lived there: {(Winnipeg, Chantal Kreviazuk), (Vancouver, Michael J. Fox), (Calgary, Stephen Harper), (Regina, Steve Nash), (Vancouver, Pamela Anderson)} So, the relation a function. c) A relation that associates a number with its double: {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} So, the relation a function. 2. Identify the domain and range of each relation in question 1. a) The domain is: The range is: b) The domain is: The range is: c) The domain is: The range is: 3. This table shows Alberta s speeding fines for different speeds in a 60 km/h zone. Speed, s (km/h) Fine, f ($) a) Why is this relation also a function? b) Identify the dependent variable and the independent variable. The depends on. So, is the dependent variable and is the independent variable. c) Write the domain and range. The domain is: The range is: Copyright 2011 Pearson Canada Inc. 263

8 4. Write in function notation. a) P 2s 15 b) y 3x 5 5. Write as an equation in two variables. a) d(t) 4t 7 b) g(x) 2x 3 d When an equation has the form y, we use symbols such as f(x), g(x), or h(x), to name the function. 6. For the function P(n) 7n 18, find: a) P(3) b) P(8) P(n) 7n 18 P(3) 7( ) 18 P(3) 18 P(3) P(8) 7. Patty lifts weights at the local gym. The equation M 5n 2.5 represents the mass lifted, M kilograms, when the number of 5-kg masses on the bar is n. a) Describe the function. Write the equation using function notation. is a function of. In function notation: b) Find the value of M(6). M(n) 5n 2.5 Substitute: n M(6) is the value of when. This means that when there are 5-kg masses on the bar, Patty lifts kg. c) Find the value of n when M(n) M(n) 5n 2.5 Substitute: n M(n) 42.5 when n This means that when there are 5-kg masses on the bar, Patty lifts kg Copyright 2011 Pearson Canada Inc.