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The function \( f(x) = -\sqrt{x} \) is an example of a negative square root function. When dealing with square roots, it is important to remember that the square root of a number is its value when raised to the power of 0.5. The negative square root means we are taking the square root of a number and then multiplying the result by -1.
This changes the direction of the graph on a standard Cartesian plane. Whereas the positive square root of \( x \), \( \sqrt{x} \), would increase as \( x \) increases, the negative square root, \( -\sqrt{x} \), decreases. This type of function is only defined for non-negative numbers since you cannot take a square root of a negative number in the set of real numbers.
This means that the domain of this function is \( x \geq 0 \). For \( f(x) = -\sqrt{x} \), every input \( x \) produces an output \( y \) that is non-positive, as the result of multiplying by the negative sign creates values that are less than or equal to zero.

Interval analysis helps us understand the behavior of a function over particular areas of its domain. For the function \( f(x) = -\sqrt{x} \), the focus is on the interval \((0, \infty)\).
This means we are interested in the behavior of this function starting just above zero and continuing indefinitely towards infinity. We check how the function behaves—not in isolation—but over specified sections.

• For small values of \( x \) approaching zero, \( -\sqrt{x} \) will start close to zero, as the square root of small numbers is also small, and the negative sign keeps it non-positive.
• As \( x \) increases to values like 1, 4, and 9, \( \sqrt{x} \) progresses to 1, 2, and 3, but with the negative in the equation, \( f(x) \) will be -1, -2, and -3 respectively.

By analyzing the interval, it is clear that the function decreases as \( x \) increases throughout our interval.

Understanding the behavior of \( f(x) = -\sqrt{x} \) gives insight into how the function acts as \( x \) progresses. We look at the general direction of the graph and other distinctive patterns.
For this function, the behavior showcases a decrease as we move from left to right on the interval \((0, \infty)\). This is a direct outcome of how the function is constructed: the square root component \( \sqrt{x} \) will naturally produce larger numbers when \( x \) becomes larger.
However, the negative sign adjoint to the square root in \( -\sqrt{x} \) ensures the function moves downwards. Thus, the increase of \( \sqrt{x} \) counterintuitively results in a decrease of \( f(x) \). The behavior is crucial in comprehending how negative transformations affect regular square root graphs and how mathematical signs influence directional changes.

A function is defined as decreasing if, for any two points \( x_1 \) and \( x_2 \) within the domain, if \( x_1 < x_2 \), then \( f(x_1) > f(x_2) \). The function \( f(x) = -\sqrt{x} \) fits this definition perfectly over the interval \((0, \infty)\).
As \( x \) progresses from small numbers to larger numbers, the outputs lessen which results in a decreasing behavior rather than increasing.

• This means as you plug in larger and larger \( x \) values into \( -\sqrt{x} \), the more negative or less positive the outcome becomes.
• For example, substituting \( x = 1 \) gives \( f(x) = -1 \) and for \( x = 4 \), \( f(x) = -2 \). These decreasing results showcase the function's decreasing nature.

Recognizing \( f(x) = -\sqrt{x} \) as a decreasing function helps students predict outcomes and understand function movement within specified intervals.