Functions and how many solutions

Is it a function, is it linear, how many solutions does it have, and reading function notation — the questions asked about a relation rather than solved from it.

5 sheets · 50 questions · one answer key · 7 pages

Recommended for grades 8–9

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Function? Sheet 1 of 5

Is it a functionSets of ordered pairs

Name: Date:

Does each set of ordered pairs show a function? Show how you know.

1(3, 12)(7, 20)(17, 31)(19, 18)(6, 20)

2(1, 16)(16, 59)(13, 56)(16, 49)

3(3, 44)(11, 44)(15, 9)(1, 47)

4(22, 45)(24, 60)(9, 18)(2, 2)

5(10, 54)(12, 47)(13, 33)(3, 57)(3, 59)

6(18, 49)(4, 52)(11, 54)(18, 2)(20, 43)

7(4, 59)(4, 1)(13, 21)(6, 33)

8(14, 59)(5, 44)(4, 13)(23, 40)(9, 38)

9(12, 28)(1, 46)(17, 18)(17, 4)

10(7, 41)(18, 52)(10, 26)(1, 26)(8, 56)

Linear or nonlinearFrom the equation

Name: Date:

Is each equation linear or nonlinear? Circle one.

1y = 35/xLinearNonlinear

2y = 10x² + 14LinearNonlinear

3y = 94LinearNonlinear

4y = 3x² + 12LinearNonlinear

5y = 60LinearNonlinear

6y = 35LinearNonlinear

7y = 16/xLinearNonlinear

8y = 26/xLinearNonlinear

9y = x/9 + 16LinearNonlinear

10y = 8x + 37LinearNonlinear

How many solutionsCompare both sides

Name: Date:

How many solutions does each equation have? Show how you know.

117x + 25 = 25 + 17x

2−15x − 39 = −39 − 15x

315x + 44 = 20x − 8

4−14x − 39 = −13x − 54

514x − 45 = 14x − 46

613x − 27 = −27 + 13x

7−6x − 38 = −11x − 52

8−8x − 5 = −8x

94x − 19 = 4x − 11

1011x − 14 = 4x + 60

Function notationEvaluating f(x)

Name: Date:

Work out the value each function asks for.

1f(x) = 7x − 10f(−5) =

2f(x) = 4x − 13f(−3) =

3f(x) = 8x − 5f(5) =

4f(x) = −9x + 20f(−3) =

5f(x) = 3x − 1f(−3) =

6f(x) = −6x + 7f(2) =

7f(x) = x + 7f(−6) =

8f(x) = x + 16f(2) =

9f(x) = −5x + 6f(5) =

10f(x) = 4x − 13f(4) =

Function notationComposing f(g(x))

Name: Date:

Work the inside function first, then the outside one.

1f(x) = 5x − 9g(x) = x + 1f(g(−3)) =

2f(x) = 2x − 8g(x) = x − 8f(g(3)) =

3f(x) = −4x − 6g(x) = 3x + 2f(g(−3)) =

4f(x) = 2x − 5g(x) = −x − 7f(g(−4)) =

5f(x) = −3x + 9g(x) = −x − 6f(g(−4)) =

6f(x) = 5x − 5g(x) = 3x − 1f(g(4)) =

7f(x) = 4x + 1g(x) = −2x + 4f(g(5)) =

8f(x) = −5x − 5g(x) = 2x − 8f(g(−2)) =

9f(x) = 4x + 8g(x) = 4xf(g(−2)) =

10f(x) = −2x + 4g(x) = −2xf(g(5)) =

Power Pack · Functions and how many solutionsAnswer key

Function?

1. (3, 12) (7, 20) (17, 31) (19, 18) (6, 20): Yes — every input appears once — 20 turns up twice, but that is an output, and two inputs may land on the same one2. (1, 16) (16, 59) (13, 56) (16, 49): No — the input 16 goes to 59 and also to 49, and one input may only go to one place3. (3, 44) (11, 44) (15, 9) (1, 47): Yes — every input appears once — 44 turns up twice, but that is an output, and two inputs may land on the same one4. (22, 45) (24, 60) (9, 18) (2, 2): Yes — every input appears once, so no input can go to two places5. (10, 54) (12, 47) (13, 33) (3, 57) (3, 59): No — the input 3 goes to 57 and also to 59, and one input may only go to one place6. (18, 49) (4, 52) (11, 54) (18, 2) (20, 43): No — the input 18 goes to 49 and also to 2, and one input may only go to one place7. (4, 59) (4, 1) (13, 21) (6, 33): No — the input 4 goes to 59 and also to 1, and one input may only go to one place8. (14, 59) (5, 44) (4, 13) (23, 40) (9, 38): Yes — every input appears once, so no input can go to two places9. (12, 28) (1, 46) (17, 18) (17, 4): No — the input 17 goes to 18 and also to 4, and one input may only go to one place10. (7, 41) (18, 52) (10, 26) (1, 26) (8, 56): Yes — every input appears once — 26 turns up twice, but that is an output, and two inputs may land on the same one

Linear?

1. y = 35/x: Nonlinear — x is underneath, so doubling x halves y instead of adding the same amount again2. y = 10x² + 14: Nonlinear — x is squared, so the steps in y grow as x grows3. y = 94: Linear — y is always 94 whatever x does — that graph is a flat line, and flat is a slope of zero rather than no slope4. y = 3x² + 12: Nonlinear — x is squared, so the steps in y grow as x grows5. y = 60: Linear — y is always 60 whatever x does — that graph is a flat line, and flat is a slope of zero rather than no slope6. y = 35: Linear — y is always 35 whatever x does — that graph is a flat line, and flat is a slope of zero rather than no slope7. y = 16/x: Nonlinear — x is underneath, so doubling x halves y instead of adding the same amount again8. y = 26/x: Nonlinear — x is underneath, so doubling x halves y instead of adding the same amount again

Power Pack · Functions and how many solutionsAnswer key

Linear?

9. y = x/9 + 16: Linear — dividing by 9 is multiplying by a number, so the steps in y stay equal10. y = 8x + 37: Linear — x is only multiplied by a number and added to, so equal steps in x give equal steps in y

Solution count

1. 17x + 25 = 25 + 17x: infinitely many2. −15x − 39 = −39 − 15x: infinitely many3. 15x + 44 = 20x − 8: one solution4. −14x − 39 = −13x − 54: one solution5. 14x − 45 = 14x − 46: no solution6. 13x − 27 = −27 + 13x: infinitely many7. −6x − 38 = −11x − 52: one solution8. −8x − 5 = −8x: no solution9. 4x − 19 = 4x − 11: no solution10. 11x − 14 = 4x + 60: one solution

Evaluating f(x)

1. f(x) = 7x − 10 f(−5) = −452. f(x) = 4x − 13 f(−3) = −253. f(x) = 8x − 5 f(5) = 354. f(x) = −9x + 20 f(−3) = 475. f(x) = 3x − 1 f(−3) = −106. f(x) = −6x + 7 f(2) = −57. f(x) = x + 7 f(−6) = 18. f(x) = x + 16 f(2) = 189. f(x) = −5x + 6 f(5) = −1910. f(x) = 4x − 13 f(4) = 3

Composing functions

1. f(x) = 5x − 9 g(x) = x + 1 f(g(−3)) = −192. f(x) = 2x − 8 g(x) = x − 8 f(g(3)) = −183. f(x) = −4x − 6 g(x) = 3x + 2 f(g(−3)) = 224. f(x) = 2x − 5 g(x) = −x − 7 f(g(−4)) = −115. f(x) = −3x + 9 g(x) = −x − 6 f(g(−4)) = 156. f(x) = 5x − 5 g(x) = 3x − 1 f(g(4)) = 507. f(x) = 4x + 1 g(x) = −2x + 4 f(g(5)) = −238. f(x) = −5x − 5 g(x) = 2x − 8 f(g(−2)) = 559. f(x) = 4x + 8 g(x) = 4x f(g(−2)) = −2410. f(x) = −2x + 4 g(x) = −2x f(g(5)) = 24

Functions and how many solutions

What is in the pack

The algebra track

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