state the domain, the range, and the intervals on which function is increasing, decreasing, or constant in interval notation

Answer:

• domain (-∞, ∞)
• range (-∞, 4]
• increasing (-∞, 0)
• decreasing (0, ∞)
• constant (only at x=0, not on any interval)

Step-by-step explanation:

The graph is of the equation y = -x^2 +4. It is a polynomial of even degree, so has a domain of all real numbers: (-∞, ∞).

The vertical extent of the graph includes y=4 and all numbers less than that:

  range: (-∞, 4]

The graph is increasing to the left of its vertex at x=0, decreasing to the right.

  increasing (-∞, 0); decreasing (0, ∞)

There is no interval on which the function is constant. It has a horizontal tangent at x=0, but a single point does not constitute an interval.

Final answer:

The domain of a function refers to all possible inputs while the range comprises all potential outputs. The function increases, decreases, or remains constant when the respective slope is positive, negative, or zero. I've provided an explanation based on the indication of the respective slopes described in your problem.

Explanation:

To determine the domain, range, and intervals of increase, decrease, or constant for a function, we need to examine the specific input and output values as well as the curvature of the function.

Domain of a function refers to all possible input values (x-values). For example, in the probability distribution function (PDF), the domain may include all numerical values or could be expressed through a non-numerical set such as different hair colors. From the provided information, I can deduce that the domain of X is {English, Mathematics, ...} - a list of all majors offered at the university, indicating all the possible inputs of this function. The domain of Y and Z are numerical, from zero up to an upper limit.

Range of a function is all the potential output values (y-values). The range is usually derived from the domain values after undergoing certain transformations via the function. Unfortunately, without further specifics about the function, I can't provide a conclusive range.

For intervals of increase, decrease, or constant, you look at the slope of the function. A function is increasing on an interval if the y-value increases as the x-value increases. Contrary to this, a function is decreasing on an interval if the y-value decreases as the x-value increases. If the y-value remains constant as the x-value varies, the function is constant on that interval. Different parts of your provided solutions indicate the function starts with positive slope (increasing), then levels off (becomes constant).

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