Question

what are the domain and range of the function on the graph? the domain includes all integers, and the range is y ≥ 0. the domain includes all integers, and the range is y > 0. the domain includes all real numbers, and the range is y ≥ 0. the domain includes all real numbers, and the range is y > 0.

Explanation:

Step1: Analyze the domain

The graph of the function (appears to be an exponential or similar curve) extends horizontally for all real - number values of \(x\) (there are no breaks or restrictions on the \(x\) - values it can take). So the domain includes all real numbers, not just integers. This eliminates the first two options which state the domain includes all integers.

Step2: Analyze the range

Looking at the \(y\) - values of the graph, the curve is above the \(x\) - axis ( \(y = 0\) line) and as \(x\) approaches negative infinity, the \(y\) - value approaches \(0\) (but never actually reaches \(0\)). So the range of the function is \(y>0\) (since \(y\) can get arbitrarily close to \(0\) but not equal to \(0\), and all positive real numbers are included). The third option has a range of \(y\geq0\) which is incorrect.

Answer:

The domain includes all real numbers, and the range is \(y > 0\).