Relations and Functions: practice solutions, Grade 9 – download the PDF

1 Domain and range of a relation ★★★

The first coordinates give the domain: \(\{-3, 0, 4, 7\}\).

The second coordinates are \(2, 5, 5, -1\). Listing each value once in increasing order, the range is \(\{-1, 2, 5\}\).

2 Function or not? ★★★

1. Each input \(1, 2, 3\) appears once: \(A\) is a function.
2. The input \(0\) is paired with \(4\) and with \(6\): \(B\) is not a function.
3. The inputs are all different. The output \(1\) repeats, which is allowed: \(C\) is a function.

3 Evaluate a linear function ★★★

\(f(2) = 4(2) + 3 = 11\).

\(f(0) = 4(0) + 3 = 3\).

\(f(-3) = 4(-3) + 3 = -12 + 3 = -9\).

4 Complete a table ★★★

\(h(-2) = -2 - 7 = -9\), \(h(0) = -7\), \(h(5) = -2\), \(h(10) = 3\).

5 Read a mapping diagram ★★★

1. \(\{(1, 4), (2, 5), (2, 7), (3, 6)\}\).
2. Domain \(\{1, 2, 3\}\); range \(\{4, 5, 6, 7\}\).
3. No. The input \(2\) has two arrows, to \(5\) and to \(7\), so it is not a function.

6 Taxi fare ★★★

The fare is the flat fee plus \(2\) dollars for each mile: \(f(m) = 2m + 3\).

\(f(6) = 2(6) + 3 = 15\). A 6-mile trip costs 15 dollars. Since a trip cannot have negative length, the domain is \(m \ge 0\).

7 Next terms of a sequence ★★★

\(d = 12 - 7 = 5\). The next terms are \(22, 27, 32\).

\(a_n = 7 + 5(n - 1) = 5n + 2\). Check: \(a_3 = 17\).

8 Vertical line test on a graph ★★★

The dashed line \(x = 3\) meets the curve at two points, \((3, 2)\) and \((3, -2)\). The test fails, so the graph is not a function. The input \(3\) has the two outputs \(2\) and \(-2\).

9 Evaluating a quadratic ★★★

\(g(0) = 1\).

\(g(3) = 9 - 12 + 1 = -2\).

\(g(-1) = 1 + 4 + 1 = 6\).

\(g(5) = 25 - 20 + 1 = 6\).

Notice that \(g(-1) = g(5)\): two different inputs, same output, still a function.

10 Solve f(x) = k ★★★

1. \(3x - 5 = 19\), \(3x = 24\), \(x = 8\).
2. \(3x - 5 = -8\), \(3x = -3\), \(x = -1\).
3. \(3x - 5 = 0\), \(3x = 5\), \(x = \dfrac{5}{3}\).

11 Domain and range from a table ★★★

Each \(x\) appears once, so it is a function. The domain is \(\{0, 1, 2, 3, 4\}\) and the range is \(\{0, 3, 6\}\) (the repeated outputs are listed once).

12 Reading a graph ★★★

The curve is \(f(x) = x^2 - 2x - 3\).

1. \(f(0) = -3\).
2. \(f(1) = -4\), the lowest point.
3. The curve crosses the \(x\)-axis at \(x = -1\) and \(x = 3\).
4. \(f(x) = 5\) at \(x = -2\) and \(x = 4\), since \(f(-2) = 4 + 4 - 3 = 5\) and \(f(4) = 16 - 8 - 3 = 5\).

13 Find the rule from a table ★★★

Each time \(x\) goes up by \(1\), \(f(x)\) goes up by \(3\), so \(f(x) = 3x + b\). With \(f(1) = 7\): \(3 + b = 7\), \(b = 4\). So \(f(x) = 3x + 4\) (check: \(f(4) = 16\)).

\(f(20) = 60 + 4 = 64\).

14 Saving money ★★★

\(a_n = 15 + 6(n - 1) = 6n + 9\).

\(a_{12} = 72 + 9 = 81\), so 81 dollars in week 12.

Solve \(6n + 9 > 100\): \(n > 15.17\), so \(n = 16\). Check: \(a_{15} = 99\) and \(a_{16} = 105\). The first week above 100 dollars is week 16.

15 Temperature function ★★★

\(F(0) = 32\); \(F(100) = 180 + 32 = 212\); \(F(-40) = -72 + 32 = -40\).

Solve \(1.8c + 32 = 50\): \(1.8c = 18\), \(c = 10\). So \(50^\circ\text{F}\) is \(10^\circ\text{C}\).

16 True or false? ★★★

1. True. A function is a relation with the extra rule of one output per input.
2. False. \(\{(1, 2), (1, 3)\}\) is a relation but not a function.
3. False. For \(f(x) = x^2\), \(f(-2) = f(2) = 4\).
4. False. It is a vertical line, so the vertical line \(x = 4\) meets it at every one of its points: the input \(4\) has infinitely many outputs.
5. True. It is a horizontal line; every vertical line meets it exactly once.

17 Evaluating with expressions ★★★

(a) \(f(a + 1) = 2(a + 1) - 3 = 2a - 1\).

(b) \(f(2x) = 2(2x) - 3 = 4x - 3\).

(c) \(f(1) = -1\), so \(f(x) + f(1) = 2x - 3 - 1 = 2x - 4\).

Since \(f(x + 1) = 2x - 1 \ne 2x - 4\), they are not equal: a function does not “distribute” over addition.

18 Fencing a garden ★★★

1. \(A(w) = w(20 - w)\).
2. \(A(5) = 5 \cdot 15 = 75\), \(A(10) = 100\), \(A(15) = 15 \cdot 5 = 75\) (square feet).
3. Both sides must be positive: \(w > 0\) and \(20 - w > 0\), so the domain is \(0 < w < 20\). Writing \(A(w) = 100 - (w - 10)^2\), the largest area is \(100\) ft\(^2\) at \(w = 10\), and the area gets close to but never reaches \(0\).

19 Two sequences race ★★★

1. \(a_n = 3 + 7(n - 1) = 7n - 4\); \(b_n = 40 - 5(n - 1) = 45 - 5n\).
2. \(a_4 = 24 < b_4 = 25\), but \(a_5 = 31 > b_5 = 20\). Sequence \(a\) passes \(b\) at \(n = 5\).
3. \(7n - 4 = 45 - 5n\) gives \(12n = 49\), \(n = \dfrac{49}{12} \approx 4.08\), not a whole number. So the sequences are never equal.

20 Find the missing constant ★★★

\(f(3) = 3a + 5 = 14\), so \(3a = 9\) and \(a = 3\).

Then \(f(x) = 3x + 5\) and \(f(-4) = -12 + 5 = -7\).

21 Far terms of a sequence ★★★

\(d = \dfrac{42 - 17}{9 - 4} = 5\). Then \(a_1 = 17 - 3 \cdot 5 = 2\) and \(a_n = 2 + 5(n - 1) = 5n - 3\).

\(a_{50} = 250 - 3 = 247\).

\(5n - 3 = 500\) gives \(n = 100.6\): not a whole number, so \(500\) is not a term. \(5n - 3 = 502\) gives \(n = 101\): \(502\) is the \(101\)st term.

22 Rule from a graph ★★★

Slope \(= \dfrac{6 - (-2)}{4 - 0} = 2\); the \(y\)-intercept is \(-2\). So \(f(x) = 2x - 2\).

\(f(10) = 20 - 2 = 18\).

\(2x - 2 = -10\) gives \(2x = -8\), \(x = -4\).