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Math 10C

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Relations and functions

Ten questions of mixed difficulty, covering Relations and functions. Print it, or work through it on screen — the answer key starts on its own page.

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Relations and functions

Math 10C · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Of these, the graph that is a function is

    1. a) a circle
    2. b) a vertical line
    3. c) a sideways parabola
    4. d) a parabola opening upward

  2. The vertical line test says that a graph is a function if

    1. a) it crosses the y-axis once
    2. b) no vertical line crosses it more than once
    3. c) it is continuous
    4. d) no horizontal line crosses it more than once

  3. The domain of a relation is

    1. a) the y-intercept
    2. b) the set of allowed input values
    3. c) the set of output values
    4. d) the slope

  4. For the graph of a line with no restrictions, the domain is

    1. a) x ≥ 0
    2. b) the y-intercept
    3. c) only whole numbers
    4. d) all real numbers

  5. A relation shows the cost of buying n tickets at $12 each. A sensible domain is

    1. a) all real numbers
    2. b) negative numbers only
    3. c) whole numbers from 0 upward
    4. d) numbers between 0 and 1

  6. Of these, the relation that is <b>not</b> a function is

    1. a) {(1, 5), (2, 5), (3, 5)}
    2. b) {(1, 2), (1, 3), (2, 4)}
    3. c) {(0, 0), (1, 1)}
    4. d) {(1, 2), (2, 3), (3, 4)}

  7. The range of the relation graphed as a parabola with vertex (2, −3) opening upward is

    1. a) y ≤ −3
    2. b) x ≥ 2
    3. c) y ≥ −3
    4. d) all real numbers

  8. If f(x) = 2x + 5, then f(x + 1) is

    1. a) 2x + 7
    2. b) 3x + 5
    3. c) 2x + 5 + 1
    4. d) 2x + 6

  9. If f(x) = x² + 1 and f(a) = 10, then a could be

    1. a) 9
    2. b) ±√10
    3. c) 3 or −3
    4. d) 10

  10. A relation is a function when

    1. a) each input has exactly one output
    2. b) each output has exactly one input
    3. c) the graph passes through the origin
    4. d) the graph is a straight line

Answer key · Relations and functions

Math 10C · maddyhelps.com

  1. d) a parabola opening upward — An upward parabola passes the vertical line test. A circle and a sideways parabola both fail it, and a vertical line fails it as badly as possible — it is one input with infinitely many outputs.
  2. b) no vertical line crosses it more than once — A vertical line is a single x-value, so two crossings would mean one input with two outputs. The horizontal line test is a different test, for whether a function has an inverse.
  3. b) the set of allowed input values — Domain is the x-values, range is the y-values. Alphabetical order is the mnemonic that works: d before r, x before y.
  4. d) all real numbers — A line continues forever in both directions, so every x-value works. The exception is a horizontal line, which is still all real numbers for domain but has a single value for range.
  5. c) whole numbers from 0 upward — You cannot buy half a ticket or a negative number of them. Domain in a word problem is set by the situation, not by the algebra, and saying so is usually worth a mark.
  6. b) {(1, 2), (1, 3), (2, 4)} — The input 1 appears twice with two different outputs, so it fails. The third set repeats an <i>output</i>, which is perfectly allowed — every input still has exactly one output.
  7. c) y ≥ −3 — The vertex is the lowest point, so no y-value below −3 is reached, and every value above it is. Range describes y; the x ≥ 2 answer is describing the wrong axis.
  8. a) 2x + 7 — Replace every x with (x + 1): 2(x + 1) + 5 = 2x + 2 + 5 = 2x + 7. The bracket matters — the 2 multiplies the whole input, not just the x.
  9. c) 3 or −3 — Set x² + 1 = 10, so x² = 9 and x = ±3. Both work, because squaring removes the sign — giving only 3 loses half the answer.
  10. a) each input has exactly one output — Inputs may not be shared out; outputs may. Two different x-values can give the same y and it is still a function — the rule is about x, not y.