Introduction to Functions

What is a Function?

A function is a special relationship between inputs and outputs where each input has exactly ONE output.

Think of a function as a machine:

• You put in an input (x)
• The machine processes it
• You get exactly one output (y)

Real-life examples:

• Vending machine: insert money (input) → get item (output)
• Temperature conversion: Fahrenheit (input) → Celsius (output)
• Square function: number (input) → its square (output)

Function Notation

Instead of writing y = 2x + 3, we can write:

f(x) = 2x + 3

Read as: "f of x equals 2x plus 3"

Parts:

• f is the name of the function (can be any letter)
• x is the input variable (independent variable)
• f(x) is the output value (dependent variable)
• f(x) replaces y

Other function names:

• g(x), h(x), F(x), etc.

Evaluating Functions

Evaluating a function means finding the output for a specific input.

Example 1: If f(x) = 3x + 5, find f(2)

Substitute x = 2: f(2) = 3(2) + 5 f(2) = 6 + 5 f(2) = 11

This means when x = 2, the output is 11.

Example 2: If g(x) = x² - 4, find g(-3)

g(-3) = (-3)² - 4 g(-3) = 9 - 4 g(-3) = 5

Example 3: If h(x) = 2x² + x - 1, find h(0)

h(0) = 2(0)² + 0 - 1 h(0) = -1

Example 4: If f(x) = x² + 3x, find f(a)

f(a) = a² + 3a

We just replace x with a!

Evaluating with Expressions

Example 1: If f(x) = x² + 1, find f(x + 2)

Substitute (x + 2) for every x: f(x + 2) = (x + 2)² + 1 f(x + 2) = x² + 4x + 4 + 1 f(x + 2) = x² + 4x + 5

Example 2: If g(x) = 2x - 3, find g(3a)

g(3a) = 2(3a) - 3 g(3a) = 6a - 3

Example 3: If h(x) = x², find h(x + h)

h(x + h) = (x + h)² h(x + h) = x² + 2xh + h²

Function vs. Not a Function

The Vertical Line Test: A graph represents a function if NO vertical line crosses it more than once.

Example: Functions

• y = 2x + 1 (linear) ✓
• y = x² (parabola) ✓
• y = |x| (absolute value) ✓

Example: Not Functions

• x² + y² = 1 (circle) ✗
• x = y² (sideways parabola) ✗

These fail because one x-value gives multiple y-values.

Functions from Tables

A table represents a function if each input appears only ONCE.

Function: ✓ | x | y | |---|---| | 1 | 3 | | 2 | 5 | | 3 | 7 |

Each x has exactly one y.

Not a function: ✗ | x | y | |---|---| | 1 | 3 | | 1 | 5 | | 2 | 7 |

x = 1 has two different outputs!

Functions from Ordered Pairs

A set of ordered pairs is a function if no two pairs have the same first element with different second elements.

Function: {(1, 2), (2, 4), (3, 6)} ✓

Not a function: {(1, 2), (1, 3), (2, 5)} ✗ Because (1, 2) and (1, 3) have same input, different outputs.

Functions from Mappings

A mapping diagram shows inputs pointing to outputs.

Function: Each input has exactly ONE arrow going out.

Not a function: If any input has MULTIPLE arrows going to different outputs.

Independent and Dependent Variables

Independent variable (input):

• The variable you choose freely
• Usually x
• Plotted on horizontal axis

Dependent variable (output):

• The variable that depends on input
• Usually y or f(x)
• Plotted on vertical axis

Example: In f(x) = 2x + 3

• x is independent (you choose x)
• f(x) is dependent (value depends on what x you chose)

Types of Functions

Linear Function: f(x) = mx + b Graph is a straight line Example: f(x) = 2x + 3

Quadratic Function: f(x) = ax² + bx + c Graph is a parabola Example: f(x) = x² - 4

Absolute Value Function: f(x) = |x| Graph is V-shaped Example: f(x) = |x - 2|

Constant Function: f(x) = c (always same output) Graph is horizontal line Example: f(x) = 5

Identity Function: f(x) = x (output equals input) Graph is line through origin with slope 1

Finding Inputs from Outputs

Sometimes we know f(x) and need to find x.

Example 1: If f(x) = 3x - 2 and f(x) = 7, find x

Set up equation: 3x - 2 = 7 3x = 9 x = 3

So f(3) = 7

Example 2: If g(x) = x² and g(x) = 16, find x

x² = 16 x = ±4

Both x = 4 and x = -4 give output 16.

Example 3: If h(x) = 2x + 1 and h(x) = -3, find x

2x + 1 = -3 2x = -4 x = -2

Domain and Range (Introduction)

Domain: The set of all possible INPUT values (x-values)

Range: The set of all possible OUTPUT values (y-values or f(x)-values)

Example 1: f(x) = x + 5 Domain: all real numbers Range: all real numbers

Example 2: g(x) = x² Domain: all real numbers Range: y ≥ 0 (squares are never negative)

Example 3: h(x) = 1/x Domain: all real numbers except 0 (can't divide by 0) Range: all real numbers except 0

Real-World Function Examples

Example 1: Temperature Conversion C(F) = (5/9)(F - 32)

Input: Fahrenheit temperature Output: Celsius temperature

C(32) = (5/9)(32 - 32) = 0°C C(212) = (5/9)(212 - 32) = 100°C

Example 2: Cost Function C(n) = 50 + 10n

Input: number of items (n) Output: total cost in dollars

C(5) = 50 + 10(5) = $100 This means 5 items cost$100.

Example 3: Distance d(t) = 60t

Input: time in hours Output: distance in miles (at 60 mph)

d(3) = 60(3) = 180 miles After 3 hours, you've traveled 180 miles.

Function Composition (Preview)

Composition means putting one function inside another.

Notation: (f ∘ g)(x) = f(g(x))

Read as: "f of g of x"

Example: If f(x) = x + 1 and g(x) = 2x

f(g(x)) = f(2x) = 2x + 1

First apply g, then apply f to the result.

Why Functions Matter

Functions are everywhere in mathematics and real life:

• Physics: distance, velocity, acceleration
• Economics: cost, revenue, profit
• Biology: population growth
• Engineering: stress, strain, load
• Computer Science: algorithms, programs
• Everyday: recipes, directions, conversions

Understanding functions is key to advanced mathematics!

Common Mistakes to Avoid

1. Confusing f(x) with f · x f(x) is NOT f times x, it's function notation

2. Adding instead of substituting f(2) means replace x with 2, not add 2

3. Not using parentheses If f(x) = x², then f(2x) = (2x)² = 4x², not 2x²

4. Thinking every relation is a function Remember: each input needs exactly ONE output

5. Mixing up domain and range Domain = inputs (x), Range = outputs (y)

Checking if an Equation Represents a Function

Method 1: Solve for y If you can solve for y and get only one y for each x, it's a function.

Example: x + y = 5 Solve: y = 5 - x ✓ Function (one y for each x)

Example: x² + y² = 25 Solve: y = ±√(25 - x²) ✗ Not a function (two y-values)

Method 2: Graph and use vertical line test

Method 3: Make a table and check for repeated inputs

Function Notation Advantages

Why use f(x) instead of y?

1. Names multiple functions clearly f(x), g(x), h(x) vs. y₁, y₂, y₃

2. Shows input explicitly f(3) clearly means input is 3

3. Easier composition f(g(x)) is clearer than nested y's

4. Better for applications C(n), d(t), P(x) tell you what the variables mean

Practice Problems Approach

When evaluating f(a):

1. Write out the function
2. Replace every x with a (use parentheses!)
3. Simplify following order of operations
4. Write final answer

When finding x given f(x):

1. Set f(x) equal to given value
2. Solve the equation for x
3. Check your answer by substituting back

Quick Reference

Function Definition: Each input has exactly ONE output

Notation: f(x) = expression

Evaluation: f(a) means substitute a for x

Vertical Line Test: Function if no vertical line crosses graph twice

Domain: All possible inputs

Range: All possible outputs

Practice Strategy

Level 1: Simple evaluation

• f(x) = x + 3, find f(5)
• g(x) = 2x, find g(7)

Level 2: Quadratic functions

• f(x) = x², find f(-3)
• h(x) = x² + 2x, find h(4)

Level 3: Expressions as inputs

• f(x) = x + 1, find f(2x)
• g(x) = x², find g(x + 1)

Level 4: Working backwards

• f(x) = 3x - 2, f(x) = 10, find x

Level 5: Applications

• Real-world function problems

Tips for Success

• Always use parentheses when substituting
• Follow order of operations carefully
• Check if relations are functions before evaluating
• Remember f(x) is just another way to write y
• Practice with different function names (f, g, h)
• Understand the input-output relationship
• Use the vertical line test on graphs
• Master evaluation before moving to composition