Question

what are the domain and range of the function on the graph?

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

Explanation:

Identify the function type

The graph shows a continuous, smooth curve that increases rapidly as \(x\) increases. This represents a continuous exponential growth function.

Determine the domain

The curve extends infinitely to the left (towards negative infinity) and continues upward and to the right (towards positive infinity) without any breaks, holes, or jumps. Since the curve is continuous and defined for all real values of \(x\), the domain includes all real numbers.

Determine the range

The curve approaches the \(x\)-axis (\(y = 0\)) as an asymptote on the left but never actually touches or crosses it. As \(x\) increases, \(y\) increases toward positive infinity. Therefore, the function values are strictly greater than zero, meaning the range is \(y > 0\).

Match with options

We combine our findings:

• Domain: all real numbers
• Range: \(y > 0\)

This matches the fourth option.

Answer:

• The domain includes all integers, and the range is \(y \ge 0\).
• The domain includes all integers, and the range is \(y > 0\).
• The domain includes all real numbers, and the range is \(y \ge 0\).
• The domain includes all real numbers, and the range is \(y > 0\). (Correct answer)