Question

graph the line -3y - 4x = 0. 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 \(-3y - 4x=0\). We want to solve for \(y\) in terms of \(x\). First, we add \(4x\) to both sides of the equation: \(-3y=4x\). Then, we divide both sides by \(- 3\) to get \(y =-\frac{4}{3}x\). This is a linear equation in the form \(y = mx + b\) where \(m =-\frac{4}{3}\) (the slope) and \(b = 0\) (the y - intercept).

Step2: Determine the domain

For a linear function of the form \(y = mx + b\) (where \(m\) and \(b\) are real numbers), there are no restrictions on the values that \(x\) can take. \(x\) can be any real number. So, the domain is all real numbers, which in interval notation is \((-\infty,\infty)\).

Step3: Determine the range

Since for every real number \(x\) we can find a corresponding real number \(y\) (by using the formula \(y=-\frac{4}{3}x\)), and as \(x\) takes on all real values, \(y\) will also take on all real values. So, the range is also all real numbers, which in interval notation is \((-\infty,\infty)\).

Answer:

The domain of the function \(y =-\frac{4}{3}x\) (which is the line \(-3y - 4x = 0\)) is \(\boldsymbol{(-\infty,\infty)}\) and the range is \(\boldsymbol{(-\infty,\infty)}\).