Introduction to Functions: practice solutions, Grade 8 – download the PDF

1 Function or not? ★★★

1. Each input 1, 2, 3, 4 appears once, so each has exactly one output. The inputs 1 and 3 share the output 4, which is allowed. It is a function.
2. The input 2 is paired with both 1 and 5. It is not a function.

2 Running a machine ★★★

\(x = 0\): \(4(0) - 3 = -3\).
\(x = 2\): \(8 - 3 = 5\).
\(x = 5\): \(20 - 3 = 17\).
\(x = -1\): \(-4 - 3 = -7\).

3 Completing a table ★★★

\(x = -1\): \(2 + 6 = 8\). \(x = 0\): \(6\). \(x = 1\): \(-2 + 6 = 4\). \(x = 3\): \(-6 + 6 = 0\).

4 Rate and initial value ★★★

1. Rate 5, initial value 12.
2. Rate \(-3\), initial value 9.
3. Rate 0.5, initial value 0 (no constant term).

5 True or false? ★★★

1. False. A function gives exactly one output for each input.
2. True. For example \(y = x^2\) gives 4 for both \(x = 2\) and \(x = -2\).
3. False. The points \((0, 0), (1, 1), (2, 4)\) are not aligned; the graph is a curve.

6 Is the change constant? ★★★

The changes in \(y\) are \(10 - 7 = 3\), \(13 - 10 = 3\), \(16 - 13 = 3\). The change is constant, so the function is linear with rate of change 3. The initial value is 7, so \(y = 3x + 7\).

7 A circle on the grid ★★★

A vertical line at \(x = 3\) meets the circle at \((3, 4)\) and at \((3, -4)\). By the vertical line test, the circle is not the graph of a function: the input 3 has two outputs.

8 Taxi fare ★★★

1. \(C = 2m + 3\).
2. \(C = 2(6) + 3 = 15\). The ride costs 15 dollars.
3. \(2m + 3 = 27 \Rightarrow 2m = 24 \Rightarrow m = 12\). The ride is 12 miles long (about 19.3 km).

9 Equation from two points ★★★

Rate: \(\dfrac{19 - 7}{6 - 2} = \dfrac{12}{4} = 3\).
Going from \(x = 2\) back to \(x = 0\) lowers \(y\) by \(3 \times 2 = 6\): initial value \(7 - 6 = 1\).
Equation: \(y = 3x + 1\). Check with \((6, 19)\): \(3(6) + 1 = 19\). Correct.

10 Linear or nonlinear? ★★★

1. Linear (rate 3, initial value \(-5\)).
2. Nonlinear: \(x\) is squared; the graph is a curve.
3. Linear: a horizontal line with rate of change 0.
4. Nonlinear: \(x\) is cubed.

11 Which function is greater? ★★★

1. For \(f\), the change is constant, \(+3\), so the rate is 3 and the initial value is 2: \(f(x) = 3x + 2\). For \(g\), the rate is 4 and the initial value is \(-1\).
2. \(f(5) = 3(5) + 2 = 17\) and \(g(5) = 4(5) - 1 = 19\). The function \(g\) is greater at \(x = 5\). (They are equal at \(x = 3\), where both equal 11.)

12 Draining a tank ★★★

1. \(V = 80 - 5t\), or \(V = -5t + 80\).
2. The rate \(-5\) means the volume drops by 5 gallons each minute. The initial value 80 is the volume at \(t = 0\).
3. \(80 - 5t = 0 \Rightarrow t = 16\). The tank is empty after 16 minutes.

13 Phone battery ★★★

1. Rate: \(\dfrac{84 - 100}{2 - 0} = -8\) percentage points per hour. Initial value 100. Equation: \(y = 100 - 8x\). Check at \(x = 4\): \(100 - 32 = 68\).
2. \(100 - 8x = 0 \Rightarrow x = 12.5\). The battery is empty after 12.5 hours.

14 Temperature conversion ★★★

1. \(1.8(0) + 32 = 32\) °F. \(1.8(25) + 32 = 45 + 32 = 77\) °F. \(1.8(100) + 32 = 212\) °F.
2. The rate is 1.8 °F per °C and the initial value is 32 °F. The equation has the form \(y = mx + b\), so the function is linear.

15 Cyclist’s trip ★★★

First stage: \(12 \times 0.5 = 6\) miles, so the graph goes from \((0, 0)\) to \((0.5, 6)\). Rest: the distance stays at 6, so the graph is flat from \((0.5, 6)\) to \((1, 6)\) (rate 0). Last stage: \(8 \times 1 = 8\) more miles, so the graph reaches \((2, 14)\). The graph changes direction at \((0.5, 6)\) and \((1, 6)\).

The cyclist covers 14 miles (about 22.5 km) in 2 hours.

16 A table that fails ★★★

The input 2 appears twice, with outputs 5 and 6, so the student is wrong: the table does not show a function. Changing the second output 6 into 5 gives the pairs \((1, 3), (2, 5), (2, 5), (3, 8)\), where each input has exactly one output. A repeated identical pair is fine.

17 Two membership plans ★★★

1. Month 3: A costs \(12 + 54 = 66\), B costs \(40 + 30 = 70\), so A is cheaper. Month 4: A costs \(12 + 72 = 84\), B costs \(40 + 40 = 80\), so B is cheaper.
2. \(18x - 10x = 40 - 12 \Rightarrow 8x = 28 \Rightarrow x = 3.5\). After 3.5 months the plans cost the same (75 dollars). Before that A is cheaper; after that B is cheaper, because B has the smaller rate (10 versus 18).

18 Error analysis ★★★

The outputs are 1, 4, 9, 16. The changes are \(3, 5, 7\). A constant rate of change would give equal changes. Increasing is not the same as increasing by a constant amount, so \(y = x^2\) is nonlinear.

19 Equation from a graph ★★★

1. Initial value \(-2\). Rate \(\dfrac{6 - (-2)}{4 - 0} = 2\). Equation: \(y = 2x - 2\).
2. Set \(y = 0\): \(2x - 2 = 0 \Rightarrow x = 1\). The graph crosses the \(x\)-axis at \((1, 0)\).

20 A burning candle ★★★

1. \(h = 12 - 1.5t\).
2. \(h = 12 - 7.5 = 4.5\) inches (about 11.4 cm).
3. The candle cannot have a negative height. \(12 - 1.5t = 0\) gives \(t = 8\). The function makes sense for \(0 \le t \le 8\).

21 Area of a square ★★★

1. Each side length gives exactly one area, so \(A\) is a function of \(s\). It is nonlinear because \(s\) is squared.
2. Side 3 gives 9 square inches and side 6 gives 36 square inches. When the side doubles, the area is multiplied by 4 (\(36 = 4 \times 9\)).

22 Missing coefficients ★★★

\(b = f(0) = 5\). Then \(4a + 5 = 17 \Rightarrow a = 3\). So \(f(x) = 3x + 5\). Then \(f(10) = 35\). For \(f(x) = 50\): \(3x + 5 = 50 \Rightarrow x = 15\).