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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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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 f(2)f(2) means ff times 2

Remediation: Emphasize that parentheses in function notation don't mean multiplication. Use machine analogy: f(2)f(2) is 'the output when 2 goes into machine ff'.

Believing that all equations with xx and yy are functions

Remediation: Show counterexamples like x=y2x = y^2 or x2+y2=1x^2 + y^2 = 1. Apply vertical line test to demonstrate multiple yy-values for one xx.

Confusing domain restrictions with the function rule

Remediation: Clarify that domain tells us WHERE the function works, not HOW it works. A restricted domain doesn't change the formula, just limits valid inputs.

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 f(x)f(x) to denote the output of function ff when the input is xx.
Key Function Concepts:
Function Notation:
f(x)=expression involving xf(x) = \text{expression involving } x
For example, if f(x)=2x+3f(x) = 2x + 3, then:
  • f(1)=2(1)+3=5f(1) = 2(1) + 3 = 5
  • f(−2)=2(−2)+3=−1f(-2) = 2(-2) + 3 = -1
Domain and Range:
  • Domain: All possible input values (xx-values)
  • Range: All possible output values (yy-values or f(x)f(x)-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 f(x)=x2−3x+2f(x) = x^2 - 3x + 2, find f(4)f(4) and f(−1)f(-1).
1

Find f(4)f(4) by substituting x=4x = 4

f(4)=(4)2−3(4)+2f(4) = (4)^2 - 3(4) + 2 → f(4)=16−12+2=6f(4) = 16 - 12 + 2 = 6
2

Find f(−1)f(-1) by substituting x=−1x = -1

f(−1)=(−1)2−3(−1)+2f(-1) = (-1)^2 - 3(-1) + 2 → f(−1)=1+3+2=6f(-1) = 1 + 3 + 2 = 6
3

Verify the calculations

Both inputs give output 6, which is valid → f(4)=6f(4) = 6 and f(−1)=6f(-1) = 6

Common Mistakes

Confusing f(x+2)f(x + 2) with f(x)+2f(x) + 2

Why it's wrong: f(x+2)f(x + 2) means substitute (x+2)(x + 2) into the function, while f(x)+2f(x) + 2 means add 2 to the output.

Correct: For f(x)=x2f(x) = x^2: f(x+2)=(x+2)2=x2+4x+4f(x + 2) = (x + 2)^2 = x^2 + 4x + 4, but f(x)+2=x2+2f(x) + 2 = x^2 + 2. Very different!

Reversing the order of composition: thinking (f∘g)(x)=g(f(x))(f \circ g)(x) = g(f(x))

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

Correct: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)): Apply gg first, then apply ff 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 f(x)=x2f(x) = x^2 with domain [0,3][0, 3], the range is [0,9][0, 9], not [0,∞)[0, \infty) (the range over all real numbers).

Assuming all relations are functions

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

Correct: x2+y2=4x^2 + y^2 = 4 (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 s(t)=12gt2s(t) = \frac{1}{2}gt^2 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 a temperature to a second scale and then to a third chains two functions: a function composition.

Example:

If C(t)=59(t−32)C(t) = \frac{5}{9}(t - 32) converts Fahrenheit to Celsius, then C(68)=59(68−32)=59(36)=20C(68) = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20 degrees Celsius.

1Try It Yourself

A weather app converts Kelvin to Celsius using C(k)=k−273C(k) = k - 273 and Celsius to Fahrenheit using F(c)=95c+32F(c) = \frac{9}{5}c + 32.

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

Step 1: Write the mathematical expression

Compute F(C(300))F(C(300)): first convert 300 K to Celsius, then to Fahrenheit (a calculation or its value):

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: C(w)=5C(w) = 5 for w≤1w \leq 1, C(w)=5+3(w−1)C(w) = 5 + 3(w - 1) for w>1w > 1.

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: f(x)=yf(x) = y
  • 2Domain is the set of all valid inputs; Range is the set of all possible outputs
  • 3Function operations: (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x), (f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x)
  • 4Composition: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)) - 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, f(x)=x2f(x) = x^2 gives f(2)=f(−2)=4f(2) = f(-2) = 4. The rule is that each input has ONE output, but different inputs can share the same output.

What's the difference between f(x)f(x) and ff?

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

Why does order matter in composition?

Because you're applying transformations in sequence. If ff doubles and gg adds 3: f(g(2))=f(5)=10f(g(2)) = f(5) = 10, but g(f(2))=g(4)=7g(f(2)) = g(4) = 7. 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 f(x)f(x) to represent the output of function ff for input xx
Composition
Applying one function to the output of another: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))
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

More in This Topic

Composite Functions

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Inverse Functions

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