Definition of a Function
What is a function?
A function is essentially an equation which assigns an input to a unique output. Functions are often represented using the notation f(x) , where x is the input and f(x) is the output.
Take a look at the two graphs shown in Figure 1. Which one is a function and which one is not?
Well, the red graph is a function since each x-value corresponds to one unique y-value. However, the blue graph is not a function since one x-value can give multiple y-values.
Let’s take a look at the red graph again. Look familiar? The equation for this graph is y = x². In function notation, this would be f(x) = x².
Note:
An important characteristic of a function is that a single value of x gives a unique value for y.
Input and Output
Let’s say for example we have a function that is defined as f(x) = x² + 3x + 5. What is the value of f(2)? We can rephrase the question by saying: “What is the output of the given function f, if the input is 2″.
So, all we need to do is substitute 2 into the function as follows:
f(2) = (2)² + 3(2) + 5
f(2) = 4 + 6 + 5
f(2) = 15
Now, here are a few examples for you to try.
[latex] \bf{Given: f(x)=\frac{1}{2}(x+3), find \hspace{0.1cm}f(1),\hspace{0.1cm}f(4),\hspace{0.1cm}f(9).}[/latex]
[latex]\bf{Given: H(x)=5x^3-2,\hspace{0.1cm} find \hspace{0.1cm} H(3)}[/latex]
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