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

1 Domain and range of a relation ★★★

Let \(R = \{(-3, 2), (0, 5), (4, 5), (7, -1)\}\). List the domain and the range of \(R\).

2 Function or not? ★★★

Decide whether each relation is a function. Explain.

1. \(A = \{(1, 3), (2, 5), (3, 7)\}\)
2. \(B = \{(0, 4), (0, 6), (1, 8)\}\)
3. \(C = \{(2, 1), (4, 1), (6, 1)\}\)

3 Evaluate a linear function ★★★

Let \(f(x) = 4x + 3\). Find \(f(2)\), \(f(0)\), and \(f(-3)\).

4 Complete a table ★★★

Complete the table for \(h(x) = x - 7\).

5 Read a mapping diagram ★★★

The diagram shows a relation from inputs to outputs.

1. Write the relation as ordered pairs.
2. Give its domain and range.
3. Is it a function?

6 Taxi fare ★★★

A taxi charges a flat fee of 3 dollars plus 2 dollars per mile. Write a function \(f(m)\) for the fare of a trip of \(m\) miles, and find the fare for 6 miles.

7 Next terms of a sequence ★★★

Consider the sequence \(7, 12, 17, \dots\). Find the common difference, the next three terms, and the explicit rule \(a_n\).

8 Vertical line test on a graph ★★★

The graph below is a sideways parabola.

Does it pass the vertical line test? Is it the graph of a function? Name the two outputs for \(x = 3\).

9 Evaluating a quadratic ★★★

Let \(g(x) = x^2 - 4x + 1\). Find \(g(0)\), \(g(3)\), \(g(-1)\), and \(g(5)\).

10 Solve f(x) = k ★★★

Let \(f(x) = 3x - 5\). Find \(x\) when (a) \(f(x) = 19\), (b) \(f(x) = -8\), (c) \(f(x) = 0\).

11 Domain and range from a table ★★★

A scientist records the values below.

Is \(y\) a function of \(x\)? Give the domain and the range.

12 Reading a graph ★★★

The graph shows a function \(f\).

Read: (a) \(f(0)\), (b) \(f(1)\), (c) the \(x\)-values where \(f(x) = 0\), (d) the \(x\)-values where \(f(x) = 5\).

13 Find the rule from a table ★★★

A linear function \(f\) has this table.

Write the rule \(f(x)\), then find \(f(20)\).

14 Saving money ★★★

Mia saves 15 dollars in week 1 and adds 6 dollars more each week, so her weekly deposits form an arithmetic sequence. Write \(a_n\), find the deposit in week 12, and find the first week when her deposit is above 100 dollars.

15 Temperature function ★★★

The function \(F(c) = 1.8c + 32\) converts degrees Celsius to degrees Fahrenheit. Find \(F(0)\), \(F(100)\), \(F(-40)\), and the Celsius temperature that gives \(50^\circ\text{F}\).

16 True or false? ★★★

True or false? Justify each answer with a short reason or a counterexample.

1. Every function is a relation.
2. Every relation is a function.
3. In a function, two different inputs can never have the same output.
4. The graph of \(x = 4\) is the graph of a function.
5. The graph of \(y = 7\) is the graph of a function.

17 Evaluating with expressions ★★★

Let \(f(x) = 2x - 3\). Simplify (a) \(f(a + 1)\), (b) \(f(2x)\), (c) \(f(x) + f(1)\). Is \(f(x + 1)\) equal to \(f(x) + f(1)\)?

18 Fencing a garden ★★★

A rectangular garden has a perimeter of 40 feet and a width of \(w\) feet. Its length is \(20 - w\) feet.

1. Write the area as a function \(A(w)\).
2. Find \(A(5)\), \(A(10)\), \(A(15)\).
3. Give a reasonable domain, and explain why \(0 < A(w) \le 100\).

19 Two sequences race ★★★

Sequence \(a\) starts at \(3\) and adds \(7\) each step. Sequence \(b\) starts at \(40\) and subtracts \(5\) each step.

1. Write \(a_n\) and \(b_n\).
2. Compare \(a_4\) with \(b_4\), and \(a_5\) with \(b_5\).
3. Is there a position \(n\) where \(a_n = b_n\)?

20 Find the missing constant ★★★

The function \(f(x) = ax + 5\) satisfies \(f(3) = 14\). Find \(a\), then compute \(f(-4)\).

21 Far terms of a sequence ★★★

An arithmetic sequence has \(a_4 = 17\) and \(a_9 = 42\). Find \(d\), \(a_1\), and \(a_n\). Then find \(a_{50}\) and decide whether \(500\) or \(502\) is a term.

22 Rule from a graph ★★★

The line shown is the graph of a linear function \(f\) through \(P(0, -2)\) and \(Q(4, 6)\).

Find the slope, the rule \(f(x)\), the value \(f(10)\), and the input for which \(f(x) = -10\).