Question

In Exercises 5-12, (a) identify the domain and range and (b) sketch the graph of the function. $$y=-\sqrt{-x}$$

Short answer

The domain of the function is \(x \leq 0\) (-∞, 0] and the range of the function is \(y \leq 0\) (-∞, 0]. The graph is a curve in the second quadrant, originating at the origin and running infinitely to the left, without crossing either axis.

Step-by-step solution

Identify the Domain

The domain of a function is the set of all real numbers that will not cause you to take square root of a negative number in the equation. For a square root function, the number that we are taking the square root of (the radicand) should be greater or equal to zero. Given that the function is \(y = -\sqrt{-x}\), this means \(-x \geq 0\). Solving this inequality for x gives \(x \leq 0\). So the domain of the function is \(x \leq 0\) or it can be written in interval notation as (-∞, 0].

Identify the Range

The range of a function is the set of all possible outputs or y-values of the function. For square root function, the output of square root is always non-negative, but because of the negative sign in front of the square root in this function, the output will always be non-positive (less than or equal to zero). Therefore, the range of the function is \(y \leq 0\) or in interval notation it is (-∞, 0].

Sketch the Graph

To sketch the graph of the function, recognize that it is similar to the graph of \(y = \sqrt{x}\) but reflected in the y-axis and inverted due to the negative signs. Start at the origin (0,0) and draw the curve in the second quadrant, running infinitely to the left. The curve should never cross the x-axis (y = 0) and the y-axis (x = 0), as these corresponds to the limits of the domain and range, respectively.

Key concepts

Domain of a Function

The concept of the domain of a function is fundamental in mathematics as it specifies the set of all possible input values (often represented by x) for which the function is defined. When working with square root functions, like our example function, \( y = -\sqrt{-x} \), it's crucial to ensure that the expression under the square root, known as the radicand, is non-negative since the square root of a negative number is not a real number. The presence of the negative inside the square root in our function indicates that the graph only exists when \( x \) is less than or equal to zero because this will result in a non-negative radicand. Consequently, the domain is all real numbers less than or equal to zero, expressed in interval notation as \((-\infty, 0]\).

Range of a Function

The range, in contrast to the domain, reflects all the possible output values (represented by y) that the function can produce. It's derived from the domain and the rules of the function itself. In our square root function \( y = -\sqrt{-x} \), the square root usually results in a non-negative value. However, due to the negative sign before the square root, all output values are flipped on the number line, producing non-positive values. Thus, for \( y = -\sqrt{-x} \), the range includes all real numbers that are less than or equal to zero, symbolized in interval notation as \((-\infty, 0]\). This tells us that the graph will never venture above the x-axis.

Sketching Graphs of Functions

A critical skill in understanding functions is sketching their graphs based on the domain and range. The graph of any function is a visual representation that displays how the output y changes with each input x. With our function, \( y = -\sqrt{-x} \), we know from its domain and range that the graph lives entirely in the second quadrant of the coordinate system, where both x and y are non-positive. We start sketching at the origin since \( (0, 0) \) is the point where the domain and range intersect, and extend the curve leftwards, making sure it approaches but never touches the y-axis, corresponding to the limit of the domain. The curve should resemble a half-parabola flipped downwards due to the negative sign in front of the square root.

Inequalities in Algebra

The study of inequalities forms an essential part of algebra and solving problems related to functions. Inequalities are relations that indicate that one expression is not equal to, but less than or greater than another. When determining the domain of \( y = -\sqrt{-x} \), we addressed the inequality \( -x \geq 0 \) to find when the square root would be of a non-negative number. Solving such inequalities frequently involves flipping the inequality sign when multiplying or dividing by a negative number. However, for our function's inequality, we simply rewrote it as \( x \leq 0 \), maintaining the direction of the inequality arrow because we are just moving the negative sign to the other side. Understanding how to manipulate and solve inequalities is vital for correctly finding the domain and range of functions.