Functions Practice

Practice understanding functions as input-output relationships. Functions help students connect equations, tables, graphs, and real-world situations.

Skill 1: Understanding Input and Output

A function connects each input to exactly one output. The input is usually called x, and the output is usually called y or f(x). Function notation f(x) means “the value of the function at x.”

Worked Example

Problem: If f(x) = 2x + 3, find f(4).

Step 1: The input is x = 4.

Step 2: Substitute 4 into the function.

f(4) = 2(4) + 3

Step 3: Simplify.

f(4) = 8 + 3 = 11

Answer: f(4) = 11

Try These

  1. If f(x) = 3x + 1, find f(2).
  2. If g(x) = x + 7, find g(5).
  3. If h(x) = 2x - 4, find h(6).
  4. If p(x) = x², find p(3).
  5. If q(x) = 10 - x, find q(4).

Answer Key

  1. f(2) = 7
  2. g(5) = 12
  3. h(6) = 8
  4. p(3) = 9
  5. q(4) = 6

Mastery Check

You are ready to move on when you can identify the input, substitute it into the function rule, and find the output.

Skill 2: Identifying Whether a Relation Is a Function

A relation is a function if each input has exactly one output. The same input cannot lead to two different outputs. If one input has more than one output, the relation is not a function.

Worked Example

Problem: Is this relation a function?

2 → 5

2 → 7

Step 1: Look at the input values.

The input 2 appears twice.

Step 2: Check the output values.

The input 2 gives two different outputs: 5 and 7.

Answer: This is not a function because one input has two different outputs.

Try These

  1. Is this a function? 1 → 4, 2 → 5, 3 → 6
  2. Is this a function? 2 → 7, 2 → 9
  3. Is this a function? 4 → 8, 5 → 8, 6 → 8
  4. Is this a function? 3 → 1, 3 → 1, 4 → 2
  5. Is this a function? 5 → 10, 5 → 12

Answer Key

  1. Yes. Each input has one output.
  2. No. The input 2 has two different outputs.
  3. Yes. Different inputs can have the same output.
  4. Yes. The input 3 repeats, but it gives the same output both times.
  5. No. The input 5 has two different outputs.

Mastery Check

You are ready to move on when you can check whether each input has exactly one output.

Skill 3: Evaluating Functions from Tables

A function table shows inputs and outputs. To evaluate a function from a table, find the given input value and read the matching output value.

Worked Example

Problem: Use the table to find f(3).

x 1 2 3 4
f(x) 5 7 9 11

Step 1: Find x = 3 in the table.

Step 2: Look directly below x = 3.

The matching output is 9.

Answer: f(3) = 9

Try These

Use the table below:

x 0 1 2 3 4
f(x) 2 5 8 11 14
  1. Find f(0).
  2. Find f(1).
  3. Find f(2).
  4. Find f(3).
  5. Find f(4).

Answer Key

  1. f(0) = 2
  2. f(1) = 5
  3. f(2) = 8
  4. f(3) = 11
  5. f(4) = 14

Mastery Check

You are ready to move on when you can find an input value in a table and identify its matching output.

Skill 4: Function Rules from Real-World Situations

Many real-world situations can be written as functions. The input is the value that changes first, and the output depends on that input. Look for the rate or pattern, then write a rule.

Worked Example

Problem: A machine produces 40 parts per hour. Write a function for the number of parts P after h hours.

Step 1: Identify the input.

The input is time in hours, h.

Step 2: Identify the output.

The output is the number of parts, P(h).

Step 3: Use the rate.

The machine produces 40 parts each hour.

Answer: P(h) = 40h

Try These

  1. A car travels 60 miles per hour. Write a function for distance d after h hours.
  2. A worker earns $18 per hour. Write a function for earnings E after h hours.
  3. A printer prints 25 pages per minute. Write a function for pages P after m minutes.
  4. A tank fills at 5 liters per minute. Write a function for volume V after t minutes.
  5. A student saves $20 each week. Write a function for savings S after w weeks.

Answer Key

  1. d(h) = 60h
  2. E(h) = 18h
  3. P(m) = 25m
  4. V(t) = 5t
  5. S(w) = 20w

Mastery Check

You are ready to move on when you can identify the input, identify the output, and write a function rule using the given rate.

Skill 5: Linear vs. Nonlinear Functions

A linear function changes at a constant rate and makes a straight line when graphed. A nonlinear function does not change at a constant rate and usually makes a curve. Recognizing the difference helps students prepare for graphs, quadratics, rates of change, and calculus.

Worked Example

Problem: Is f(x) = 3x + 2 linear or nonlinear?

Step 1: Look at the form of the function.

The variable x has an exponent of 1.

Step 2: Check the pattern.

A function in the form y = mx + b changes at a constant rate.

Answer: f(x) = 3x + 2 is linear.

Try These

  1. Is f(x) = 2x + 5 linear or nonlinear?
  2. Is g(x) = x² nonlinear or linear?
  3. Is h(x) = -4x + 1 linear or nonlinear?
  4. Is p(x) = x³ nonlinear or linear?
  5. Is q(x) = 7 linear or nonlinear?

Answer Key

  1. Linear
  2. Nonlinear
  3. Linear
  4. Nonlinear
  5. Linear. A constant function is a horizontal line.

Mastery Check

You are ready to move on when you can explain that linear functions have a constant rate of change and nonlinear functions do not.

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