Representing Functions
What is a Function?
A function is like a special rule that connects two sets of things. For every input you put in, the function gives you exactly one output. Think of it as a perfect vending machine: if you press button A1, you always get the same bag of chips. It never gives you a soda instead. The input is the button you press, and the output is the item you receive.
Mathematically, we often call the input $x$ and the output $y$ or $f(x)$. The most important rule is: One input, one output. If one input could give you two different outputs, then it's not a function.
The Function Machine: A Simple Analogy
One of the easiest ways to understand a function is to imagine it as a function machine. You feed a number into the machine, it follows its internal rule, and a new number pops out.
For example, consider a "double and add one" machine. If you input 3, the machine doubles it to 6 and then adds 1, giving an output of 7. If you input 5, it outputs 11. This machine represents the function $f(x) = 2x + 1$.
Six Ways to Represent a Function
There isn't just one right way to show a function. Different situations call for different representations. Being able to move between them is a key math skill.
| Representation | Description | Example | Best Used For | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Verbal Description | Describing the rule in words. | "The output is two times the input plus one." | Initial understanding, real-world scenarios. | ||||||||
| Function Machine | A diagram showing input -> process -> output. | Input: 3 -> (x2, +1) -> Output: 7 | Conceptual introduction, simple calculations. | ||||||||
| Table of Values | A list of input-output pairs. |
| Seeing patterns, discrete data points. | ||||||||
| Ordered Pairs | A set of points (input, output). | {(1, 3), (2, 5), (3, 7)} | Listing specific pairs, set notation. | ||||||||
| Graph | A visual plot on a coordinate plane. | A straight line through points (1,3), (2,5), etc. | Seeing the overall shape and behavior. | ||||||||
| Equation | A mathematical formula. | $y = 2x + 1$ or $f(x) = 2x + 1$ | General rule, precise calculations, calculus. |
From Words to Equations and Graphs
Let's see how we can take one function and represent it in multiple ways. Suppose we have a function described as: "The area of a square depends on the length of its side."
- Verbal: "The area is the side length multiplied by itself."
- Equation: If we let $s$ represent the side length and $A$ represent the area, the equation is $A = s^2$.
Table:
s (side) A (area) 1 1 2 4 3 9 - Graph: This would be a curve called a parabola that gets steeper as $s$ increases.
Functions in Action: Real-World Applications
Functions are not just abstract math concepts; they describe countless real-world relationships.
In Physics: The distance a car travels is a function of its speed and time. The equation $d = rt$ (distance = rate × time) is a function. For a fixed speed, the distance depends only on the time you travel.
In Business: The total cost of buying apples is a function of the number of pounds you buy. If apples cost $1.50 per pound, the function is $C = 1.5p$, where $p$ is the number of pounds. A graph of this would be a straight line starting at the origin.
In Science: The temperature in degrees Celsius can be converted to Fahrenheit using the function $F = \frac{9}{5}C + 32$. This is a perfect example of a function represented by an equation that is useful every day.
Choosing the Right Representation
Different representations have different strengths. If you want to know the output for a specific input, an equation or table might be best. If you want to see the overall trend or pattern, a graph is ideal. If you are explaining a relationship to someone, a verbal description or function machine might be the clearest. A good math student learns to use the representation that best fits the task at hand.
Common Mistakes and Important Questions
Q: Is a circle on a graph a function?
No. A circle fails the vertical line test. If you draw a vertical line through the right side of a circle, it will cross the graph at two points. This means that for a single x-value (input), there are two different y-values (outputs). This violates the fundamental rule of a function. However, you can describe a circle with equations, and you can break it into two separate functions: the top half of the circle and the bottom half.
Q: What is the difference between an equation and a function?
All functions can be represented by equations (but not all representations are equations). However, not all equations represent functions. An equation is just a statement that two expressions are equal, like $x^2 + y^2 = 25$ (a circle). For an equation to represent a function, it must assign exactly one output (y) to every input (x) in its domain. The equation $y = 2x + 1$ does this, so it is a function.
Q: Why is the function notation $f(x)$ used instead of just $y$?
Using $f(x)$ is more powerful and clear. First, it names the function (we could have $g(x)$, $h(x)$, etc.). Second, it explicitly shows the input variable. This is especially helpful when evaluating the function for a specific value. For example, "$f(2)$" clearly means "the output of function f when the input is 2." It's a more precise language for advanced mathematics.
Representing functions is a fundamental skill in mathematics that allows us to understand and describe relationships in the world around us. From the simple function machine to the precise algebraic equation, each representation offers a unique perspective on how inputs are transformed into outputs. The ability to translate between verbal descriptions, tables, graphs, and equations empowers us to solve problems, make predictions, and communicate ideas effectively. Remember the core principle: a function is a consistent rule that assigns exactly one output to every valid input. Mastering its many representations is key to unlocking higher mathematics and scientific reasoning.
Footnote
[1] Domain: The set of all possible input values (x-values) for a function. It defines what you are allowed to put into the function.
[2] Range: The set of all possible output values (y-values) that result from using the function. It defines what you can get out of the function.
[3] Vertical Line Test: A visual way to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function.