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Functions and Notation

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

A function assigns exactly one output to each allowed input. The notation f(3) names the output when the input is 3; it does not mean f multiplied by 3. Different inputs may share an output: f(x) = x^2 is a function even though f(-2) = f(2). To connect representations, keep the meaning and units of both variables fixed. A rule gives an output for each input in its domain; a table records selected input-output pairs; a graph places those pairs as points (input, output). A point belongs to a function's graph exactly when its coordinates satisfy the rule and its input is allowed. A few table entries can support a proposed rule, but cannot prove that this is the only rule between or beyond those entries. Before extending a table, state an assumption, such as a constant rate of change. To decide whether a relation is a function, look for a single input paired with two different outputs; repeated outputs alone cause no problem.

What you'll learn

Why it matters: A rental rule connects time with cost. Its table, graph, and equation should describe the same charges and use the same time units.

Worked example

Problem. A bicycle rental costs a $6 starting fee plus $4 per hour. Let C(t) be its cost in dollars for t hours, with 0 <= t <= 8. Write the rule, interpret C(3), and decide whether (3, 18) belongs on its graph.

  1. The fixed fee is paid once, so the cost starts at 6 dollars; 4t is the hourly charge.
  2. The rule is C(t) = 6 + 4t on the stated time interval 0 <= t <= 8.
  3. Substitute the input 3: C(3) = 6 + 4(3) = 18.
  4. C(3) = 18 means that a 3-hour rental costs 18 dollars; it does not mean 18 hours.
  5. The point (3, 18) has an allowed input and satisfies the rule, so it belongs on the graph.

Answer: C(t) = 6 + 4t; C(3) = $18; (3, 18) is on the graph.

Lesson preview: this page shows 5 practice items. The interactive activity contains 32 practice items and 17 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. For f(x) = 3x^2 - 2, compute f(-2).

Show solution
  1. Substitute -2 for the whole input: 3(-2)^2 - 2.
  2. Square first: (-2)^2 = 4.
  3. 3(4) - 2 = 10.

Answer: 10

2. Which relation is a function even though two different inputs have the same output?

Choices: {(-2, 4), (2, 4), (3, 9)} · {(1, 2), (1, 3), (2, 4)} · {(0, 0), (0, 1)} · {(3, 7), (3, -7)}

Show solution
  1. In the first relation, every input appears with exactly one output.
  2. The shared output 4 is allowed.
  3. Each other relation pairs one input with two different outputs.

Answer: {(-2, 4), (2, 4), (3, 9)}

3. A table contains (0, 1), (1, 3), and (2, 5). A learner says the output MUST be 7 when the input is 3. What is the strongest justified response?

Choices: 7 follows if a constant-rate linear rule is assumed, but the three entries alone do not force it. · 7 is forced because every table defines a linear function. · The output must be 6 because 3 is the input. · The table cannot represent a function because its outputs increase.

Show solution
  1. The listed outputs increase by 2 for each input increase of 1.
  2. The linear rule y = 2x + 1 fits those entries and would give 7.
  3. A different rule can agree at inputs 0, 1, and 2 while giving another output at 3; the linear assumption must be stated.

Answer: 7 follows if a constant-rate linear rule is assumed, but the three entries alone do not force it.

4. A delivery service uses C(n) = 5 + 2n dollars for a whole number n of packages. A customer pays $19. How many packages were delivered?

Show solution
  1. Here the output is given, so solve 5 + 2n = 19 for the input.
  2. Subtract 5: 2n = 14; divide by 2: n = 7.
  3. Seven is a permitted whole-number input, and C(7) = 19.

Answer: 7

5. If f(x) = x + 4, find f(9).

Show solution
  1. Substitute 9 for x.
  2. 9 + 4 = 13.

Answer: 13

Explain and apply

Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.

Compare representations and explain your assumption

A table gives (0, 2), (1, 5), (2, 8). Assuming a constant rate, write a rule, find the output at input 4, and describe how to plot the graph for 0 <= x <= 4. Then explain why the original three entries alone do not establish every point on that graph.

Review rubric and solution
  • Find a rate of 3 per input unit and an initial value of 2.
  • Write y = 3x + 2 and calculate y = 14 at x = 4.
  • Describe axes, plotted points, and the line segment for the specified interval.
  • State that extending between and beyond the table entries uses the constant-rate assumption.
  1. Each output increases by 3 when the input increases by 1, so the assumed linear rule is y = 3x + 2.
  2. At x = 4, y = 3(4) + 2 = 14.
  3. With x horizontal and y vertical, plot (0, 2), (1, 5), (2, 8), (3, 11), and (4, 14), then join them by the line segment from (0, 2) to (4, 14).
  4. The three original points do not specify all other outputs. The line segment follows from the additional constant-rate assumption.

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