College Algebra Corequisite

Module 5:

Function Basics

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Affirmations

• I am better when I create a healthy balance in my life
• I chose to move forward every day, growing and learning as I go!
• I surround myself with people who treat me well

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Introduction to Functions

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Learning Goals

1 Define what a function is and apply the vertical line test to identify functions

2 Use function notation to represent and evaluate functions

3 Recognize the graphs of fundamental toolkit functions

Deepen your understanding and form connections within these skills:

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Relations and Functions

• Relation: A set of ordered pairs
• Domain: Set of first components (inputs)
• Range: Set of second components (outputs)
• Function: A special relation where:
• Each input has exactly one output
• No x-value is repeated with different y-values

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Example: Relations vs Functions

This IS a function:

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(1,2), (2,4), (3,6), (4,8), (5,10)

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Each x-value pairs with exactly one y-value

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This is NOT a function:

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(1,2), (1,3), (2,4), (3,6)

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x = 1 pairs with both 2 and 3

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Vertical Line Test

Vertical Line Test: A graph represents a function if no vertical line intersects the graph more than once.

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Why it works:

• Each x-value (input) must have only one y-value (output)
• Vertical line shows all possible y-values for one x-value
• If line crosses more than once → not a function

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Try It!

Determine if each graph represents a function:

1. 2. 3.

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Function Notation

• Function notation: f(x) means "f of x"
• f(x) represents the output when x is input
• Example:

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When x = 3,

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Evaluating Functions

Steps to evaluate f(a):

1. Replace x with a
2. Follow order of operations
3. Simplify

Example: Given the function , evaluate f(2) and f(-1).

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Solution:

To find f(2):

Replace x with 2 in the function

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To find f(-1):

Replace x with -1 in the function

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Try It!

If , find:

1. f(0)

2. f(2)

3. f(-1)

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Basic Toolkit Functions

Essential functions to know:

1. Linear: f(x) = x
• Straight line through origin
• Domain & Range: all real numbers
2. Quadratic: f(x) = x²
• U-shaped parabola
• Domain: all reals, Range: [0, ∞)
3. Absolute Value: f(x) = |x|
• V-shaped graph
• Domain: all reals, Range: [0, ∞)

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Try It!

Match each function with its graph:

1. f(x) = |x| 2. f(x) = x² 3. f(x) = x

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A. B. C.

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Domain and Range

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Learning Goals

Deepen your understanding and form connections within these skills:

Determine the set of all possible input values for a function based on its equation

x.1

Identify the set of all possible inputs (domain) and outputs (range) from looking at a graph

x.2

Figure out the allowed inputs and outputs for the fundamental toolkit functions

x.3

Sketch piecewise functions, showing each segment with its own rule on the graph

x.4

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Domain and Range

Domain: All possible input values

• The "raw materials" for our function machine
• What x-values can we put into the function?
• Consider: division by zero, even roots of negatives

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Range: All possible output values

• What y-values can come out of the function?
• The "products" of our function machine
• Found by considering function behavior

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Interval Notation

Uses brackets to describe sets of numbers

• [a, b]: Includes endpoints a and b
• (a, b): Does not include endpoints
• [a, ∞): All numbers ≥ a
• (-∞, b]: All numbers ≤ b

Example:

• (0, 100] means all numbers > 0 and ≤ 100

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Finding Domain: Key Rules

• Basic Functions: Often all real numbers
• Fractions: Cannot divide by zero
• Find where denominator = 0
• Exclude those values
• Even Roots: Cannot take even root of negative
• Find where radicand < 0
• Exclude those values

Example: Find domain of:

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Solution: �

Set denominator = 0

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Exclude x = 2

Write in interval notation:

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Try It!

Find the domain of the following:

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1.

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2.

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Finding Domain & Range from Graphs

Strategies:

• Domain: All x-values (horizontal spread)
• Range: All y-values (vertical spread)
• Look for gaps, asymptotes
• Note endpoints/arrows

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On this example graph, the domain is [-5,∞) and the range is (-∞, 5].

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Domain & Range of Key Functions

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Linear:

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• Domain: (-∞, ∞)
• Range: (-∞, ∞)

Quadratic:

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• Domain: (-∞, ∞)
• Range: [0, ∞)

Square Root:

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• Domain: [0, ∞)
• Range: [0, ∞)

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Piecewise Functions

Functions defined by different rules for different parts of the domain

Example:

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Key Points:

• Each piece has its own rule
• Must specify where each piece applies
• Check boundaries carefully

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Try It!

Find domain and range from the piecewise function:

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1.

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Rates of Change and Function Behavior

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Learning Goals

1 Find the average rate of change of a function

2 Identify parts of a graph where the function is going up, going down, or staying the same.

3 Identify the highest and lowest points, both overall and at specific spots, on a graph

Deepen your understanding and form connections within these skills:

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Rate of Change

Rate of Change: How output changes compared to input change

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Average rate of change:

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Examples:

Speed: 60 miles per hour

Growth: 2 inches per year

Cost: $5 per unit

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Example: Average Rate of Change

Find the rate of gas prices over 2 years:

• 2020: $2.17/gallon
• 2022: $3.95/gallon

Solution:

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Interpretation: Gas prices increased by 89 cents per year on average

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Try It!

A tree grows from 6 feet to 15 feet over 3 years. Find:

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1. Total change in height

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2. Average rate of change

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Function Behavior

Increasing: Output increases as input increases

• Positive rate of change
• Upward slope

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Decreasing: Output decreases as input increases

• Negative rate of change
• Downward slope

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Constant: Output stays same as input changes

• Zero rate of change
• Horizontal line

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Local Extrema

Local Maximum: Higher than nearby points

• Changes from increasing to decreasing

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Local Minimum: Lower than nearby points

• Changes from decreasing to increasing

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Absolute Extrema

• Absolute Extrema: highest and lowest points over entire domain

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• Not every function has an absolute maxima or minima

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Example: Finding Extrema

For the quadratic function , verify:

1. The function has a local minimum at x = 2
2. The function is increasing for all x > 2
3. The function is decreasing for all x < 2

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To verify the minimum at x = 2:

• The function is a parabola that opens upward
• The axis of symmetry is ��
• This means x = 2 is the vertex
• Since the parabola opens upward and x = 2 is the vertex, this point is the minimum

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To verify the function increases for x > 2:

• This is a parabola opening upward
• Any x-value greater than the vertex will give a larger y-value
• Therefore, the function increases for all x > 2

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To verify the function decreases for x < 2:

• This is a parabola opening upward
• Any x-value less than the vertex will give a larger y-value
• Moving from right to left before the vertex, y-values increase
• Therefore, the function decreases for all x < 2

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Try It!

For the given function, find all:

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1. Local maxima and minima

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2. Increasing and decreasing intervals

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Tips for Analyzing Functions

• Look for where graph changes direction
• Check endpoints of domain
• Consider behavior as x approaches ±∞
• Test points in each interval
• Verify with calculations

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Real-World Applications

• Population growth rates
• Business profit trends
• Temperature changes
• Speed of vehicles
• Cost optimization

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Function Foundations: A Journey Through Representations

Embark on a mathematical journey through four interactive stations exploring the world of functions. Working in teams, you'll investigate:

• How functions behave through different representations
• The secrets of domain and range
• Patterns in rates of change
• Real-world applications that bring functions to life

Each station builds your understanding while challenging you to think deeper about how functions work in our world.

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Closing Slide

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Next Steps…..

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Questions…..

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Anything else….

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