Learn what makes a relationship a function, how to use function notation, and how to identify the domain and range of functions through real-world examples.
Understanding the fundamental building blocks of functions
Definition: A function is a relationship between two variables where each input (x) has exactly one output (y).
Think of it like a machine: you put in one number, and you get out exactly one result.
Key Rule: For every x, there is exactly one y
Standard Form: y = f(x)
Read as "y equals f of x" or "y is a function of x"
Examples:
Let's explore functions using temperature conversion between Celsius and Fahrenheit
Function: F(C) = (9/5)C + 32
Where C is temperature in Celsius and F is temperature in Fahrenheit
Examples:
Domain: All possible input values (Celsius temperatures)
In this case: all real numbers (any temperature is possible)
Range: All possible output values (Fahrenheit temperatures)
In this case: all real numbers (any Fahrenheit temperature is possible)
Function Rule: Each Celsius temperature gives exactly one Fahrenheit temperature
This is a function because every input has exactly one output
A visual way to determine if a relationship is a function
Rule: If any vertical line intersects the graph more than once, it's NOT a function.
If every vertical line intersects the graph at most once, it IS a function.
Why this works:
A vertical line represents a single x-value. If it hits the graph twice, that means one x-value has two y-values, which violates the function rule.
✓ Function: y = 2x + 1 (straight line)
Any vertical line intersects this line exactly once.
✗ Not a Function: x² + y² = 25 (circle)
A vertical line through the center intersects the circle twice.
Deepen your understanding with more complex function concepts
Linear Functions: f(x) = mx + b
Straight line graphs, constant rate of change
Quadratic Functions: f(x) = ax² + bx + c
Parabolic graphs, U-shaped curves
Exponential Functions: f(x) = aˣ
Rapid growth or decay, curved graphs
Square Root Functions: f(x) = √x
Half-parabola, only defined for x ≥ 0
Addition: (f + g)(x) = f(x) + g(x)
Add the outputs of two functions
Subtraction: (f - g)(x) = f(x) - g(x)
Subtract the outputs of two functions
Multiplication: (f · g)(x) = f(x) · g(x)
Multiply the outputs of two functions
Composition: (f ∘ g)(x) = f(g(x))
Use the output of g as input for f
Functions are everywhere in our daily lives
Cost Function: C(x) = fixed cost + variable cost
Total cost depends on number of items produced
Revenue Function: R(x) = price × quantity
Total revenue depends on sales volume
Profit Function: P(x) = R(x) - C(x)
Profit is revenue minus cost
Distance Function: d(t) = v₀t + ½at²
Distance depends on time, initial velocity, and acceleration
Temperature Conversion: F(C) = (9/5)C + 32
Fahrenheit depends on Celsius temperature
Population Growth: P(t) = P₀e^(rt)
Population depends on time and growth rate
Hash Functions: h(x) = unique identifier
Each input produces a unique output for data storage
Image Processing: f(pixel) = transformed pixel
Each pixel value is transformed to create effects
Encryption: E(message) = encrypted text
Each message is transformed into secure code
Test your understanding with these comprehensive function problems
Given: The relationship between a student's height and their shoe size.
Is this a function? Why or why not?
Answer: This is NOT a function.
Two students can have the same height but different shoe sizes. One input (height) can have multiple outputs (shoe sizes).
Given: f(x) = 3x - 2
Find f(5) and f(-1).
Solution:
f(5) = 3(5) - 2 = 15 - 2 = 13
f(-1) = 3(-1) - 2 = -3 - 2 = -5
Given: f(x) = 2x + 1 and g(x) = x² - 3
Find (f + g)(2) and (f - g)(1).
Solution:
(f + g)(2) = f(2) + g(2) = (2(2)+1) + (2²-3) = 5 + 1 = 6
(f - g)(1) = f(1) - g(1) = (2(1)+1) - (1²-3) = 3 - (-2) = 5
Given: h(x) = √(x - 2)
Find the domain and range of h(x).
Solution:
Domain: x ≥ 2 (because √(x-2) requires x-2 ≥ 0)
Range: y ≥ 0 (square root always gives non-negative results)