MATH122 Lecture 1.1

A function is like a rule that takes an input and gives you exactly one output. Think of it as a machine: you put something in, and it gives you something specific out. In business, these inputs and outputs are usually quantities, like hours worked (input) and money earned (output).

Key Terms

• Function: A rule that connects each input to only one output.

• Input (Domain): What you put into the function; the independent variable, meaning you control it (e.g., hours worked, time).

• Output (Range): What the function gives back; the dependent variable, meaning its value changes based on the input (e.g., pay, distance).

• Model: A function used to describe a real-world situation.

How Functions Are Shown (Four Ways)

A function can be described in different ways, but they all show the same relationship:

• Words: A simple sentence explaining the rule.

  • Example: "Pay equals $20$ per hour for each hour worked."

• Formula: A mathematical equation.

  • Example: $Pay = 20 \times hours$

• Graph: A picture on a coordinate plane.

  • The input (independent variable) is usually on the x-axis (horizontal).

  • The output (dependent variable) is usually on the y-axis (vertical).

  • Example: A point $(1, 20)$ on a graph means that when the input is $1$ (hour), the output is $20$ (pay).

• Table: A list of input and output pairs.

  • Example (Joe's Pay):

  • Hours (Input): $0, 1, 2, 3$

  • Pay (Output): $0, 20, 40, 60$

Real-World Examples

• Joe's Pay Function

  • Scenario: Joe earns money based on hours worked at $20$ per hour.

  • This is a model of income.

  • Input: Hours ($h$).

  • Output: Pay ($p$).

  • Formula: $p(h) = 20h$

  • Evaluation: To find Joe's pay for $5$ hours, we calculate $p(5) = 20 \times 5 = 100$. So, Joe earns $100$ for $5$ hours.

• Distance Running Graph

  • A graph can show how far someone is from home over time.

  • X-axis: Time (e.g., minutes).

  • Y-axis: Distance from home (e.g., miles).

  • By looking at the graph, you can see:

  • When the person was running fast or slow (steepness of the line).

  • When they stopped to rest (flat line, distance not changing).

  • When they were moving away from home or returning (line going up or down).

  • This graph is a model that tells a story about the runner's journey.

Why This Matters

Being able to understand and switch between these different ways of showing functions (words, formulas, graphs, tables) is essential for:

• Analyzing situations.

• Communicating information clearly.

• Applying math to real-world problems