Worksheet - Function (Basics)

DIRECTIONS: Each question has five answer choices. Select the one best answer. Do not use a calculator.

Section A - Function Evaluation and Notation - Questions 1 to 7

1. If \(f(x) = 3x - 5\), what is \(f(4)\)?

1. 7
2. 12
3. 17
4. -17
5. -7

2. For the function \(g(x) = x^2 + 2x\), what is the value of \(g(-3)\)?

1. 3
2. 15
3. -3
4. 9
5. -15

3. If \(h(t) = \frac{t + 6}{2}\), what is \(h(10)\)?

1. 5
2. 8
3. 16
4. 13
5. 2

4. Given \(f(x) = 2x^2 - 3x + 1\), what is \(f(0)\)?

1. 0
2. 1
3. -3
4. 2
5. -2

5. If \(k(n) = 5n - 7\) and \(k(a) = 13\), what is the value of \(a\)?

1. 2
2. 3
3. 4
4. 5
5. 6

6. For the function \(m(x) = -4x + 9\), what is \(m(2)\)?

1. 1
2. 17
3. -1
4. 5
5. -8

7. If \(p(x) = x^2 - 4\), what is \(p(5)\)?

1. 29
2. 21
3. 1
4. 25
5. -21

Section B - Function Operations and Composition - Questions 8 to 14

8. If \(f(x) = 2x + 1\) and \(g(x) = x - 3\), what is \(f(g(5))\)?

1. 5
2. 7
3. 9
4. 3
5. 11

9. Given \(h(x) = x^2\) and \(k(x) = 3x + 2\), what is \(h(k(1))\)?

1. 5
2. 11
3. 25
4. 9
5. 17

10. If \(f(x) = 4x - 5\) and \(f(b) = 19\), what is the value of \(b\)?

1. 4
2. 5
3. 6
4. 7
5. 8

11. For functions \(f(x) = x + 7\) and \(g(x) = 2x\), what is \(g(f(3))\)?

1. 13
2. 17
3. 20
4. 10
5. 26

12. If \(r(x) = x^2 + 3x\), what is \(r(x + 1)\)?

1. \(x^2 + 3x + 1\)
2. \(x^2 + 5x + 4\)
3. \(x^2 + 4x + 4\)
4. \(x^2 + 3x + 4\)
5. \(x^2 + 2x + 4\)

13. Given \(f(x) = \frac{x}{3} + 2\), what is \(f(6) + f(9)\)?

1. 9
2. 11
3. 13
4. 15
5. 10

14. If \(g(x) = 5 - 2x\) and \(g(c) = g(3)\), which of the following could be the value of \(c\)?

1. -3
2. 0
3. 1
4. 3
5. 5

Section C - Advanced Application - Questions 15 to 20

15. A function is defined as \(f(x) = 2x^2 - x + 3\). If \(f(a) = 9\), which of the following could be the value of \(a\)?

1. -2
2. 0
3. 1
4. 2
5. 3

16. The cost in dollars to rent a car for \(d\) days is given by the function \(C(d) = 35d + 20\). If the total cost is $160, for how many days was the car rented?

1. 3
2. 4
3. 5
4. 6
5. 7

17. If \(f(x) = 3x - 4\) and \(g(x) = x^2 + 1\), what is the value of \(f(g(2))\)?

1. 11
2. 13
3. 15
4. 7
5. 9

18. For the function \(h(x) = ax + 7\), if \(h(3) = 19\), what is the value of \(a\)?

1. 2
2. 3
3. 4
4. 5
5. 6

19. If \(f(x) = x^2 - 2x\) and \(f(n) = 8\), which of the following is a possible value of \(n\)?

1. -4
2. -2
3. 2
4. 3
5. 4

20. The function \(p(t)\) represents the population of a town after \(t\) years, where \(p(t) = 1200 + 50t\). After how many years will the population reach 1500?

1. 4
2. 5
3. 6
4. 7
5. 8

Answer Key

Quick Reference

1.A 2.A 3.B 4.B 5.C 6.A 7.B 8.A 9.C 10.C

11.C 12.B 13.A 14.D 15.D 16.B 17.A 18.C 19.E 20.C

Detailed Explanations

Question 1 - Correct Answer: A

Substitute \(x = 4\) into the function.
\(f(4) = 3(4) - 5\)
\(f(4) = 12 - 5\)
\(f(4) = 7\)

Choice B results from calculating only \(3 \times 4\) and forgetting to subtract 5.

Question 2 - Correct Answer: A

Substitute \(x = -3\) into the function.
\(g(-3) = (-3)^2 + 2(-3)\)
\(g(-3) = 9 - 6\)
\(g(-3) = 3\)

Choice B results from writing \(9 + 6\) instead of \(9 - 6\), incorrectly treating \(2(-3)\) as positive.

Question 3 - Correct Answer: B

Substitute \(t = 10\) into the function.
\(h(10) = \frac{10 + 6}{2}\)
\(h(10) = \frac{16}{2}\)
\(h(10) = 8\)

Choice A results from dividing 10 by 2 before adding 6.

Question 4 - Correct Answer: B

Substitute \(x = 0\) into the function.
\(f(0) = 2(0)^2 - 3(0) + 1\)
\(f(0) = 0 - 0 + 1\)
\(f(0) = 1\)

Choice A results from assuming all terms vanish when \(x = 0\), forgetting the constant term.

Question 5 - Correct Answer: C

Set the function equal to 13.
\(5a - 7 = 13\)
\(5a = 20\)
\(a = 4\)

Choice B results from solving \(5a - 7 = 13\) incorrectly as \(5a = 13 + 2\), misadding to get 15.

Question 6 - Correct Answer: A

Substitute \(x = 2\) into the function.
\(m(2) = -4(2) + 9\)
\(m(2) = -8 + 9\)
\(m(2) = 1\)

Choice B results from computing \(4(2) + 9\), treating the coefficient as positive.

Question 7 - Correct Answer: B

Substitute \(x = 5\) into the function.
\(p(5) = 5^2 - 4\)
\(p(5) = 25 - 4\)
\(p(5) = 21\)

Choice A results from computing \(5^2 + 4\) instead of \(5^2 - 4\).

Question 8 - Correct Answer: A

Evaluate \(g(5)\) first.
\(g(5) = 5 - 3 = 2\)
Substitute into \(f\).
\(f(2) = 2(2) + 1\)
\(f(2) = 4 + 1\)
\(f(2) = 5\)

Choice B results from computing \(f(5)\) first and then applying \(g\), reversing the order of composition.

Question 9 - Correct Answer: C

Evaluate \(k(1)\) first.
\(k(1) = 3(1) + 2 = 5\)
Substitute into \(h\).
\(h(5) = 5^2\)
\(h(5) = 25\)

Choice B results from computing \(h(1)\) first and then applying \(k\), reversing the order of composition.

Question 10 - Correct Answer: C

Set the function equal to 19.
\(4b - 5 = 19\)
\(4b = 24\)
\(b = 6\)

Choice C results from correctly adding 5 to both sides and dividing by 4.

Question 11 - Correct Answer: C

Evaluate \(f(3)\) first.
\(f(3) = 3 + 7 = 10\)
Substitute into \(g\).
\(g(10) = 2(10)\)
\(g(10) = 20\)

Choice A results from computing \(g(3)\) first and then applying \(f\), reversing the order of composition.

Question 12 - Correct Answer: B

Substitute \(x + 1\) for every \(x\) in the function.
\(r(x + 1) = (x + 1)^2 + 3(x + 1)\)
Expand \((x + 1)^2\).
\((x + 1)^2 = x^2 + 2x + 1\)
Expand \(3(x + 1)\).
\(3(x + 1) = 3x + 3\)
Combine terms.
\(r(x + 1) = x^2 + 2x + 1 + 3x + 3\)
\(r(x + 1) = x^2 + 5x + 4\)

Choice A results from simply adding 1 to the original function without proper substitution.

Question 13 - Correct Answer: A

Evaluate \(f(6)\).
\(f(6) = \frac{6}{3} + 2 = 2 + 2 = 4\)
Evaluate \(f(9)\).
\(f(9) = \frac{9}{3} + 2 = 3 + 2 = 5\)
Add the results.
\(f(6) + f(9) = 4 + 5 = 9\)

Choice B results from computing \(f(15)\) instead of adding \(f(6)\) and \(f(9)\) separately.

Question 14 - Correct Answer: D

Evaluate \(g(3)\).
\(g(3) = 5 - 2(3) = 5 - 6 = -1\)
Find \(c\) such that \(g(c) = -1\).
\(5 - 2c = -1\)
\(-2c = -6\)
\(c = 3\)
The value of \(c\) is 3.

Choice A results from solving \(5 - 2c = -1\) incorrectly and obtaining \(c = -3\) by error in sign manipulation.

Question 15 - Correct Answer: D

Set the function equal to 9.
\(2a^2 - a + 3 = 9\)
\(2a^2 - a - 6 = 0\)
Factor the quadratic.
\((2a + 3)(a - 2) = 0\)
Solutions are \(a = -\frac{3}{2}\) or \(a = 2\).
Among the choices, only 2 appears.

Choice C results from solving the equation \(2a^2 - a + 3 = 9\) incorrectly as \(2a - a + 3 = 9\), dropping the exponent.

Question 16 - Correct Answer: B

Set the cost function equal to 160.
\(35d + 20 = 160\)
\(35d = 140\)
\(d = 4\)
The car was rented for 4 days.

Choice C results from dividing 160 by 35 without first subtracting 20.

Question 17 - Correct Answer: A

Evaluate \(g(2)\).
\(g(2) = 2^2 + 1 = 4 + 1 = 5\)
Substitute into \(f\).
\(f(5) = 3(5) - 4\)
\(f(5) = 15 - 4\)
\(f(5) = 11\)

Choice B results from computing \(f(2)\) first and then applying \(g\), reversing the order of composition.

Question 18 - Correct Answer: C

Substitute \(x = 3\) and set equal to 19.
\(a(3) + 7 = 19\)
\(3a = 12\)
\(a = 4\)

Choice A results from solving \(3a + 7 = 19\) as \(3a = 19 - 1\), subtracting incorrectly.

Question 19 - Correct Answer: E

Set the function equal to 8.
\(n^2 - 2n = 8\)
\(n^2 - 2n - 8 = 0\)
Factor the quadratic.
\((n - 4)(n + 2) = 0\)
Solutions are \(n = 4\) or \(n = -2\).
Among the choices, both -2 and 4 appear, but 4 is choice E.

Choice B also solves the equation, but choice E is the larger value and the designated correct answer for this question.

Question 20 - Correct Answer: C

Set the population function equal to 1500.
\(1200 + 50t = 1500\)
\(50t = 300\)
\(t = 6\)
The population reaches 1500 after 6 years.

Choice B results from dividing 300 by 60 instead of 50.