Question

graph the line. -4x + 3y = 12. give the domain and range. use the graphing tool to graph the equation. click to enlarge graph

Explanation:

Step1: Rewrite the equation in slope - intercept form

We start with the equation \(-4x + 3y=12\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the \(y\) - intercept).
Add \(4x\) to both sides of the equation: \(3y=4x + 12\).
Then divide each term by 3: \(y=\frac{4}{3}x + 4\).

Step2: Analyze the domain of a linear function

A linear function of the form \(y = mx + b\) (where \(m=\frac{4}{3}\) and \(b = 4\) in our case) has no restrictions on the values of \(x\) that we can plug in. We can use any real number for \(x\) (positive, negative, zero, rational, irrational, etc.). So the domain is all real numbers. In interval notation, the domain is \((-\infty,\infty)\).

Step3: Analyze the range of a linear function

Since for every real number \(x\) we plug into \(y=\frac{4}{3}x + 4\), we get a real number \(y\), and as \(x\) takes on all real values, \(y\) will also take on all real values. For example, if \(x\) approaches \(\infty\), \(y=\frac{4}{3}x + 4\) approaches \(\infty\), and if \(x\) approaches \(-\infty\), \(y=\frac{4}{3}x + 4\) approaches \(-\infty\). So the range is also all real numbers. In interval notation, the range is \((-\infty,\infty)\).

Answer:

The domain of the function \(y=\frac{4}{3}x + 4\) (which represents the line \(-4x + 3y = 12\)) is \(\boldsymbol{(-\infty,\infty)}\) and the range is \(\boldsymbol{(-\infty,\infty)}\).