Review of Functions

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

Advanced25 minLesson
Printable worksheet

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.

Try it now

If f(x)=3x+2f(x) = 3x + 2, what is f(4)f(4)?

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.

Interactive Visual

Function graph

-20246-20246810xy(4, 6)(-1, 6)

Tap or hover over the graph to read off values.

Tap a name to show or hide its graph.

Function graph

-4-2024-20246810xy(-2, 4)(0, 1)(3, 7)

Tap or hover over the graph to read off values.

f(x)
  • At x = 0: from the left f(x) → 0, from the right f(x) → 1. The values differ, so the limit does not exist (jump). f(0) = 1.

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = x^2 - 3x + 2

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
If f(x)=3x+2f(x) = 3x + 2, what is f(4)f(4)?

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

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.
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.
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.
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
Today's challengeOne problem, three tries

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