Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=\sqrt{x}-2$$

Short answer

The domain of the function is \(x\geq0\) and the range is \(y\geq-2\). The graph starts from the point (0,-2) and goes upwards.

Step-by-step solution

Identify the domain

The domain, D, of the function \(y=\sqrt{x}-2\) can be determined by setting \(\sqrt{x}\) greater than or equal to 0, as the square root of a number is only defined for positive numbers and zero in the real number system. So, the domain is \(x\geq0\).

Identify the range

Now we find the range of the function. The smallest value for \(\sqrt{x}\) in this case is 0 (when x is 0). Subtracting 2 from the smallest value of \(\sqrt{x}\) gives -2. Hence, the smallest y-value is -2. As x approaches infinity, \(\sqrt{x}\) also approaches infinity and \(\sqrt{x}-2\) approaches infinity as well. Therefore, the range of the function is all y-values such that \(y\geq-2\).

Sketch the graph

Start from point (0,-2) as the square root function starts from zero and we subtract 2 from every value. Sketch a curve going upwards as \(x\) increases from \(0\) to infinity. The curve will never go below -2 on the y-axis.

Key concepts

Function Graphing

Graphing a function is an essential skill for understanding its behavior and for visualizing how the input (x-values) relate to the output (y-values). In our example, we have the function

\( y = \sqrt{x} - 2 \).

To graph this function, one must recognize it as a transformed version of the basic square root function \( y = \sqrt{x} \). The subtraction by 2 indicates a vertical shift downwards by 2 units. To start the graph:

• Plot the starting point at (0, -2), corresponding to the smallest x-value of 0.
• As x increases, the y-values follow a curve that increases at a decreasing rate. This is characteristic of the square root function.
• Continue this pattern, plotting points and connecting them in a smooth curve to illustrate the function's behavior.

With practice, graphing functions becomes intuitive, allowing for quicker and more accurate sketches that illustrate the function's domain and range clearly.

Square Root Function

The square root function is a type of radical function, expressed as \( y = \sqrt{x} \), and it exhibits specific properties that influence its graph. Here's what you should know about this function:

• It is defined only for non-negative x-values, meaning its domain is \( x \geq 0 \).
• The output values start at 0 and increase as x increases.
• The rate of increase is quick at first but slows down; graphically, this appears as a curve that starts steep and flattens out.
• Any transformation, like adding or subtracting a number, affects the position but not the basic shape of the graph.

In the case of our function \( y = \sqrt{x} - 2 \), the -2 indicates that the entire graph is shifted down by 2 units from the basic square root graph.

Mathematical Functions Analysis

Analyzing mathematical functions involves determining their characteristics, such as domain, range, intercepts, and asymptotes. For the function

\( y = \sqrt{x} - 2 \),

the analysis goes as follows:

• Domain: Since \( x \) must be non-negative for a square root to be real, the domain is \( x \geq 0 \).
• Range: The function's smallest value is at \( x = 0 \), yielding \( y = -2 \), so the range is \( y \geq -2 \).
• Intercepts: The y-intercept is at (0, -2). There is no x-intercept since the function never touches the x-axis.
• Behavior: As \( x \) approaches infinity, \( y \) also increases without bound, so the function has no horizontal asymptote. The x-axis serves as a sort of asymptotic boundary the function will not cross.

Analyzing these characteristics not only aids in understanding the function's graph but also provides insight into its application in real-world contexts.