What are Functions?
If you checked out our intro to functions lesson or the lesson on representing functions, you know that a function takes an input value and assigns it exactly one output value. Think about a function as a vending machine. You give the function an input value (the code you type in) and it assigns you an output (candy, pretzels, etc.). If you enter the same input code the next time, it should give you the exact same output that it gave you the last time.
There are several different ways to represent a function. Mapping diagrams use ovals and arrows to show how the inputs and outputs are matched up. You can also use tables, graphs, or a set of ordered pairs to represent a function.
Open the next tab to see how an equation can represent a function.
Functions as Equations
An equation can also be a function. Remember, a function takes an input value and assigns it an output value. If a function is written as an equation, the equation tells you exactly what the function is going to do with the input value to get the output value.
For example, the equation y = 3x is a function. You plug your input value in for x and it gets multiplied by 3. If your input value was 5, the output would be 3(5) = 15. If the input value was 7, the output would be 3(7) = 21.
When a function is written as an equation, the equation tells you the relationship between the input and output values. The function y = x - 1 takes input values and subtracts 1 to give the output values. The function y = x/2 takes input values and divides them by 2 to give the output values.
Open the next tab to learn about function notation.
Mathematicians often work with more than one function at a time, so it's helpful to have a way to give a function a "name." In function notation, the most popular letter to use is "f" for function, but just about any letter can be used to name a function.
Let's look at how the function y = 2x will look if we write it in function notation. Instead of writing y, we'll call it function f and write f(x) = 2x. This is pronounced "f of x." The letter f is the name of the function. The x is written in parentheses to tell you that the variable for the input values is x (it does not mean f times x).
Remember, f(x) = 2x is the exact same thing as y = 2x. Function notation just allows us to name a function (so we can know if we're talking about function f, function g, function h, etc.) and it gives us information about the input values (the x inside the parentheses tells us that we'll plug the input values in for x).
Here's what a few other functions might look like in function notation:
In the examples above, we used the letters f, g, and h to name the three different functions. These are the most common letters used to name functions, but really about any letter can be used to name a function. Notice that all three examples use the variable x as the input variable. While x is probably the most common variable you'll use in algebra, it doesn't always have to be x. If you look at the table below, you can see several different ways to write the same function using different variables.
The variable inside the parentheses is just the variable that's used for the input in the function. Remember, f(x) tells you that f is the name of the function and the x in the parentheses tells you that the function uses the variable x for the input values.
The most common mistake students make is to think that f(x) means f is being multiplied by x. This is not the case. The first letter is just the name of the function. What's inside the parentheses represents the input value. If you see g(t), that means that g is the name of the function and the variable that's used for the input is t. If you see h(a), that means h is the name of the function and a is the variable used for the input values.
Let's say you're given the function f(x) = x - 3. This means that the function has been given the name f and the input variable is x. What does this function do? It subtracts 3 from the input value to give you the output.
Let's say you're asked to find f(8). What does this mean? It does NOT mean f times 8. Remember, the first letter f tells you the name of the function. What's in parentheses represents the input value. f(8) tells us that we need to plug in 8 for the input value.
We know that f(x) = x - 3. To find f(8), we just plug in 8 for x! f(8) = 8 - 3 = 5.
Here's a few more examples in the table below. To find f(4), you just plug in 4 for the input value.
Notice in the bottom row of the table, the function had two places we needed to plug in 4. If your function has more than one term with a variable, make sure you plug your input value into each one.
Want to learn more about functions? Check out the lesson on the Vertical Line Test.
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