Teacher Guide: Review of Functions

Master the fundamentals of functions including notation, domain, range, and operations before advancing to pre-calculus.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Advanced Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Evaluate functions using function notation for various inputs
Determine the domain and range of functions from equations and graphs
Perform function operations: addition, subtraction, multiplication, and division
Compute composite functions and understand the order of composition
Evaluate piecewise functions by identifying the correct piece for each input
Apply the vertical line test to determine if a relation is a function

Prerequisites

• Understanding of algebraic expressions and equations
• Familiarity with interval notation
• Knowledge of basic function types (linear, quadratic, absolute value)
• Graphing skills on the coordinate plane

Discussion Starters

1. Why do mathematicians use function notation instead of just writing equations?
2. Can you think of real-world processes that are NOT functions? (Where one input could have multiple outputs)
3. If you compose a function with its inverse, what happens? Can you explain why?
4. How would you explain the vertical line test to someone who has never heard of functions?

Common Misconceptions

Thinking that means times 2

Believing that all equations with and are functions

Confusing domain restrictions with the function rule

Differentiation Ideas

For Struggling Students:

• Use function machines with physical manipulatives
• Start with simple linear functions before quadratics
• Provide domain and range on graphs before algebraic expressions
• Use tables of values to reinforce input-output relationships

For On-Level Students:

• Practice with polynomial, rational, and radical functions
• Find domains algebraically for various function types
• Compose two functions and verify with specific values
• Analyze piecewise functions with two or three pieces

For Advanced Students:

• Explore inverse functions and their compositions
• Find domains of composite functions
• Analyze functions with multiple domain restrictions
• Create real-world models using function composition

Standards Alignment

F-IF.A.1 (CCSS.MATH.CONTENT.HSF.IF.A.1)
Understand that a function from one set to another set assigns to each element of the domain exactly one element of the range
F-IF.A.2 (CCSS.MATH.CONTENT.HSF.IF.A.2)
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation
F-IF.B.5 (CCSS.MATH.CONTENT.HSF.IF.B.5)
Relate the domain of a function to its graph and to the quantitative relationship it describes
F-BF.A.1 (CCSS.MATH.CONTENT.HSF.BF.A.1)
Write a function that describes a relationship between two quantities

Lesson Resources

visualInteractive Function Grapher
Plot and explore various function types
activityComposition Chain
Chain multiple functions together and trace the transformations
worksheetDomain Detective
Practice finding domains of various function types

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A function is a relation where each input has exactly one output. We write to denote the output of function when the input is .

Key Function Concepts:

Function Notation:

For example, if , then:

Domain and Range:

Domain: All possible input values (-values)
Range: All possible output values (-values or -values)

Vertical Line Test: A graph represents a function if and only if every vertical line intersects the graph at most once.

Worked Examples

Given , find and .

Find by substituting

→

Find by substituting

→

Verify the calculations

Both inputs give output 6, which is valid → and

Answer: and . Note: Different inputs can produce the same output in a function.

Common Mistakes

Confusing with

Why it's wrong: means substitute into the function, while means add 2 to the output.

Correct: For : , but . Very different!

Reversing the order of composition: thinking

Why it's wrong: The notation can be confusing. The function on the right is applied first.

Correct: : Apply first, then apply to the result.

Forgetting to check domain restrictions when finding range

Why it's wrong: The range depends on which inputs are allowed. A restricted domain often restricts the range too.

Correct: For with domain , the range is , not .

Assuming all relations are functions

Why it's wrong: Not every equation or graph represents a function. The vertical line test must pass.

Correct: (a circle) is NOT a function because vertical lines can intersect it twice.

Why It Matters

Functions are the foundation of all advanced mathematics:

Physics: Motion equations like describe position as a function of time
Economics: Supply and demand curves model price relationships
Computer Science: Every program is essentially a collection of functions transforming inputs to outputs
Engineering: Transfer functions describe how systems respond to inputs
Biology: Population models use functions to predict growth over time

Mastering functions is essential for calculus, where you'll study how functions change and accumulate.

Real World Applications

Temperature Conversion

Converting between Celsius and Fahrenheit is a function composition application.

Example:

If converts Fahrenheit to Celsius, then degrees Celsius.

1Try It Yourself

A weather app converts Kelvin to Celsius using and Celsius to Fahrenheit using .

Find the temperature in Fahrenheit when it's 300 Kelvin.

Step 1: Write the mathematical expression

First convert 300K to Celsius, then to Fahrenheit:

Shipping Cost Calculator

Many businesses use piecewise functions to calculate shipping costs based on weight.

Example:

A company charges based on weight: 5 dollars for packages up to 1 kg, then 3 dollars per additional kg.

2Try It Yourself

Shipping cost function: for , for .

What's the cost to ship a 4 kg package?

Step 1: Write the mathematical expression

Since 4 > 1, use the second formula:

Key Takeaways

1A function assigns exactly one output to each input:
2Domain is the set of all valid inputs; Range is the set of all possible outputs
3Function operations: ,
4Composition: - apply the inner function first
5The vertical line test determines if a graph represents a function

Frequently Asked Questions

Can a function have the same output for different inputs?

Yes! For example, gives . The rule is that each input has ONE output, but different inputs can share the same output.

What's the difference between and ?

is the function itself (a rule or process), while is the output value when you input . Think of as a machine and as what comes out when you put in.

Why does order matter in composition?

Because you're applying transformations in sequence. If doubles and adds 3: , but . Different processes yield different results.

Glossary

Function: A relation where each input has exactly one output
Domain: The set of all possible input values for a function
Range: The set of all possible output values of a function
Function notation: Writing to represent the output of function for input
Composition: Applying one function to the output of another:
Piecewise function: A function defined by different expressions for different parts of its domain
Vertical line test: A graphical test: if any vertical line crosses a graph more than once, it's not a function