Variables in Functions

Article Contents

1. Independent Variable (Input)
2. Dependent Variable (Output)
3. Why "Dependent" and "Independent"?
4. Variables in Function Notation
5. Variables on a Graph
6. Real-Life Examples
7. Common Mistakes
8. Interactive Exercises

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1. Independent Variable (Input) {#independent-variable}

The independent variable is the value we choose freely. It is the input of the function.

• Usually denoted by $x$
• Plotted on the horizontal axis (x-axis) of a graph
• We decide its value first

Example

In the function $f(x)=3x-1$, the variable $x$ is independent. We can set $x$ to any value we like:

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2. Dependent Variable (Output) {#dependent-variable}

The dependent variable is the result -- it depends on the input. It is the output of the function.

• Usually denoted by $y$ or $f(x)$
• Plotted on the vertical axis (y-axis) of a graph
• Its value is determined by the function rule

Example

For $f(x)=3x-1$:

We chose $x$. The function determined $y$.

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3. Why "Dependent" and "Independent"? {#why-dependent}

The names tell us which variable controls the other:

• Independent ($x$): "I am free -- nobody tells me what to be."
• Dependent ($y$): "I depend on $x$ -- when $x$ changes, I change too."

Analogy

Think of a light dimmer:

• The position of the dial = independent variable (you control it)
• The brightness of the light = dependent variable (it depends on the dial)

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4. Variables in Function Notation {#variables-in-notation}

The notation $y=f(x)$ tells us clearly which variable is which:

Different Letters, Same Idea

Functions can use any letters. The argument (inside parentheses) is always independent:

The variable inside the parentheses is always the input (independent).

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5. Variables on a Graph {#variables-on-graph}

On a coordinate graph:

• The horizontal axis shows the independent variable $x$
• The vertical axis shows the dependent variable $y=f(x)$

Each point on the graph represents one input-output pair: $(x,f(x))$.

To read a function value from the graph:

1. Choose an $x$-value on the horizontal axis
2. Go vertically up (or down) to the curve
3. Read the $y$-value on the vertical axis

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6. Real-Life Examples {#real-life-examples}

Example 1: Distance and Time

A car drives at 80 km/h. The distance $d$ depends on time $t$:

• Independent: time $t$ (hours) -- it passes on its own
• Dependent: distance $d$ (km) -- changes as time goes on

Example 2: Age and Height

A child's height $h$ changes with age $a$:

• Independent: age $a$ (years)
• Dependent: height $h$ (cm)

Example 3: Workers and Time

If a job takes 120 person-hours, the time $t$ depends on the number of workers $w$:

• Independent: number of workers $w$
• Dependent: time to finish $t$ (hours)

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7. Common Mistakes {#common-mistakes}

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8. Interactive Exercises {#interactive-exercises}

Practice identifying variables:

• Identify Independent and Dependent Variables

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Summary

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Related Articles

• Introduction to Functions -- what is a function
• Functions and Tables -- representing functions with tables
• Functions and Graphs -- plotting and reading graphs
• Domain of a Function -- which inputs are allowed
• Range of a Function -- which outputs are possible