How to Identify Relations and Functions? (+FREE Worksheet!)

Relations and Functions – Example 1:

Relations and Functions – Example 2:

Relations and Functions – Example 3:

Relations and Functions – Example 4:

Solution:

Tutor-style math help

Identify Relations and Functions: what to notice and how to work it

Functions skill

A function is a rule that gives each input exactly one output. Function notation, tables, graphs, and equations are different ways to show the same input-output relationship.

What to notice first

Ask what kind of input you are given. Sometimes you substitute a number, sometimes you read a graph, and sometimes you combine two rules.

Common student mistake

Do not read \(f(4)\) as multiplication. It means the output of f when the input is 4.

Key formulas and cues

\(f(a)\text{ means replace }x\text{ with }a\)

\((f\circ g)(x)=f(g(x))\)

\(f^{-1}(x)\text{ reverses }f(x)\)

A reliable path

1. Identify the inputFind the x-value, expression, or inner function being used.
2. Apply the ruleSubstitute with parentheses so signs and powers stay clear.
3. Interpret the outputState the value, point, interval, domain, range, or inverse relationship.

Worked examples

Evaluate a function

Example: \(f(x)=4x-3\), find \(f(2)\)

1. Replace x with 2.
2. Compute 4(2) – 3.
3. Simplify.

Answer: \(5\)

Compose functions

Example: \(f(x)=x+1\), \(g(x)=2x\), find \(f(g(3))\)

1. Find g(3) = 6.
2. Use that as the input for f.
3. f(6) = 7.

Answer: \(7\)

Open pop-up practice

Try one before moving on

Try: If \(h(x)=2x^2\), find \(h(-3)\).

Answer: \(18\). Use parentheses: \(2(-3)^2=18\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

x

Identify Relations and Functions: pop-up practice

Answer these quick questions, then use the feedback to decide which part of the lesson to review.

Choose an answer to begin.

1. \(f(3)\) means:

AThe output when x = 3Bf times 3C3 divided by f

2. A function gives each input:

AExactly one outputBAt least two outputsCNo output

3. If \(f(x)=x+4\), then \(f(6)=\)

A\(10\)B\(2\)C\(24\)

The relation is not a function. Input 9 has two outputs: 15 and 22. Remember that in a function, the input value must have one and only one value for the output.

The domain is all the \(x\)-coordinates: {1, 6, 9, 11}

The domain is all the \(y\)-coordinates: {5, 15, 16, 22, 25}

Exercises for Identifying Relations and Functions

Determine if the following relations are functions. Then state the domain and range.

1) g={(1, 5), (2, 6), (3, 7), (5, 10), (6, 20), (10, 35)}

2)

3)

4) f={(-2, 3), (7, 5), (5, -4), (-2, 10), (10, 15)}

1)The relation is a function.

The domain is all the \(x\)-coordinates: {1, 2, 3, 5, 6, 10}

The domain is all the \(y\)-coordinates: {5, 6, 7, 10, 20, 35}

2)The relation is not a function. Input -10 has two outputs: -2 and 3.

The domain is all the \(x\)-coordinates: {-10, -5, 0, 5, 15}

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The domain is all the \(y\)-coordinates: {-2, 0, 5, 12, 3, 20}

3)The relation is not a function. Input 7 has two outputs: 15 and 18.

The domain is all the \(x\)-coordinates: {5, 6, 7}

The domain is all the \(y\)-coordinates: {15, 16, 17, 18}

4)The relation is not a function. Input -2 has two outputs: 3 and 10.

The domain is all the \(x\)-coordinates: {-2, 7, 5, 10}

The domain is all the \(y\)-coordinates: {3, 5, -4, 10, 15}

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