A function is a relation where each input, or x-value, corresponds to exactly one output, or y-value. To visualize this, we can use the vertical line test. If any vertical line intersects the graph at more than one point, it's not a function. Here's an example of a function, y equals x, and a non-function, x equals y squared.
Linear equations are mathematical expressions of the form y equals m x plus b, where m represents the slope and b represents the y-intercept. The slope indicates the steepness of the line, and the y-intercept is where the line crosses the y-axis. Most linear equations represent functions because each x-value corresponds to exactly one y-value. Let's look at a few examples: y equals x, y equals zero point five x plus one, and y equals negative x plus two.
Now let's analyze the specific equation y equals three x minus four. This is a linear equation in slope-intercept form, where the slope m is three and the y-intercept b is negative four. To demonstrate that this is a function, we can create a table of input-output pairs. For x values of negative two, negative one, zero, one, and two, we calculate the corresponding y values. Each x-value produces exactly one y-value, confirming it's a function.
To graph the equation y equals three x minus four, we start by plotting the y-intercept at the point zero, negative four. Then, we use the slope of three, which means a rise of three units for every one unit run. From the y-intercept, we go up three units and right one unit to find another point at one, negative one. We can also go down three units and left one unit to find the point negative one, negative seven. Connecting these points with a straight line gives us the graph of our function.
To verify that y equals three x minus four is a function, we can use three methods. First, the vertical line test: when we draw vertical lines across the graph, each line intersects the graph at exactly one point. Second, input-output analysis: for any x-value we choose, the equation produces exactly one y-value. Third, examining the domain and range: the domain is all real numbers because we can substitute any real number for x, and the range is also all real numbers because the line extends infinitely in both directions.
In conclusion, yes, y equals three x minus four is definitely a function of x. We've confirmed this through multiple methods: it's a linear equation in slope-intercept form, it passes the vertical line test, and every x-value corresponds to exactly one y-value. This specific function is a linear function. We can also express it using function notation as f of x equals three x minus four.