A function is a relation where each input (x-value) corresponds to exactly one output (y-value). For example, the line y equals x is a function because any vertical line intersects it at most once. However, a circle like x squared plus y squared equals one is not a function because a vertical line can intersect it at two points.
Let's analyze the equation y equals 3x minus 2. This is a linear equation in slope-intercept form, y equals mx plus b. Here, the slope m is 3 and the y-intercept b is -2. For any given x-value, there is exactly one corresponding y-value. For example, when x is 0, y is -2. When x is 1, y is 1. When x is -1, y is -5.
Let's visualize y equals 3x minus 2 on a coordinate plane. We start by marking the y-intercept at (0, -2). Then we plot additional points like (1, 1) and (2, 4). Connecting these points gives us a straight line. To verify this is a function, we can apply the vertical line test. Vertical lines at different x-values each intersect the graph at exactly one point, confirming this is indeed a function.
Let's systematically verify that y equals 3x minus 2 is a function. First, the domain: all real numbers are valid inputs for this equation. Second, the range: all real numbers are possible outputs. Third, there's a one-to-one correspondence between inputs and outputs. Fourth, algebraically, each x-value produces exactly one y-value. Our table of values confirms this pattern, and the graph shows a straight line where each x corresponds to exactly one y. Therefore, we can definitively conclude that y equals 3x minus 2 is indeed a function of x.
We can express the relationship using function notation. Instead of y equals 3x minus 2, we write f of x equals 3x minus 2. This notation emphasizes the input-output relationship, where f of x represents the output corresponding to input x. For example, f of 0 equals -2, f of 1 equals 1, and f of 3 equals 7. Function notation is particularly useful in mathematics as it clearly shows the dependence of the output on the input variable.