A function is a special type of relation where each input value, or x-value, corresponds to exactly one output value, or y-value. This means that for any given x, there is only one possible y. Let's look at some visual examples to understand this better.
To determine if a relation is a function from its graph, we use the vertical line test. If you can draw any vertical line that intersects the graph more than once, then the graph does not represent a function. If every vertical line intersects the graph at most once, then it is a function.
Let's analyze the specific equation y equals 3x minus 3. This is a linear equation in slope-intercept form, y equals mx plus b, where m is the slope and b is the y-intercept. For any given x-value, this equation produces exactly one y-value through substitution. For example, when x equals zero, y equals negative three; when x equals one, y equals zero; and when x equals two, y equals three.
To visually confirm that y equals 3x minus 3 is a function, let's graph it. The graph is a straight line with slope 3 and y-intercept negative three. Now we'll apply the vertical line test by drawing several vertical lines across the graph. As you can see, each vertical line intersects the graph at exactly one point, confirming that this equation represents a function.
Linear functions like y equals 3x minus 3 have specific properties. Their domain is all real numbers, meaning you can substitute any real number for x. Their range is also all real numbers, meaning they can produce any real number as output. They are one-to-one, passing the horizontal line test, and they are continuous with no breaks. All non-vertical lines represent functions, which is why our equation is definitely a function.
We can express the equation y equals 3x minus 3 using function notation as f of x equals 3x minus 3. This notation emphasizes the input-output relationship of the function. Let's evaluate the function for specific inputs: f of zero equals 3 times zero minus 3, which equals negative three; f of one equals 3 times one minus 3, which equals zero; and f of two equals 3 times two minus 3, which equals three. This shows how function notation makes the relationship between input and output explicit.
To conclude, y equals 3x minus 3 is indeed a function of x. We have multiple pieces of evidence supporting this: it passes the vertical line test, each x-value produces exactly one y-value, it can be written in function notation as f of x equals 3x minus 3, and it exhibits all the properties of a linear function. Understanding function definitions and testing methods enables us to identify functions in various mathematical contexts.