Question

In Exercises 13-20, use a grapher to (a) identify the domain and range and (b) draw the graph of the function. $$y=x^{3 / 2}$$

Short answer

The domain of \(y=x^{3 / 2}\) is \(x \geq 0\) and its range is \(y \geq 0\). The graph starts from the origin and rises in the first quadrant.

Step-by-step solution

Function Analysis

First, we recognize that this function is a power function with an exponent of \(3/2\). This means that the function only accepts non-negative real numbers as inputs. Thus, the domain of the function is \(x \geq 0\). The output or range of the function will also be non-negative real numbers since any non-negative number raised to a real number exponent results in a non-negative number. Thus, the range of the function is \(y \geq 0\).

Sketch the graph of the function

To graph the function \( y=x^{3/2} \), start with some basic \(x, y\) pairs that satisfy the function. Some basic ones given the domain are \((0,0), (1,1), (4,8)\). Plot these points. Then sketch the curve that goes through these points. The function \( y=x^{3/2} \) results in a curve that starts from the point \((0,0)\) and continues to rise in the first quadrant of the coordinate system.

Key concepts

Domain of a Function

Understanding the domain of a function is essential for correctly graphing it. The domain refers to the set of all possible input values (usually 'x') for the function. For example, when dealing with the power function \(y=x^{3/2}\), we must consider what types of numbers can be placed into the function without causing mathematical errors. Because we cannot take the square root of a negative number without involving complex numbers, which are beyond the scope of standard power functions, the domain of this function is restricted to non-negative numbers. Hence, we succinctly express the domain as \(x \geq 0\).

It's important to note that the determination of the domain depends on the kind of power within the function. Had the exponent been a whole number or an even root, the domain could have been different. This particular piece of analytical reasoning is crucial and often included in exercise improvement advice for providing a clearer path to understanding how domains are established for various functions.

Range of a Function

Parallel to understanding the domain, grasping the concept of the range of a function is just as critical. The range encompasses all possible output values (usually 'y') that the function can produce. In the context of the power function \(y=x^{3/2}\), the operation of raising a non-negative number to the power of \(3/2\) will always yield a non-negative result. Therefore, the range of this function, like its domain, includes only non-negative numbers, formally expressed as \(y \geq 0\).

A solid mastery of both domain and range not only contributes to correct graphing but also deepens the comprehension of the behavior of functions within their respective constraints. In teaching materials, highlighting the symbiotic relationship between domain and range can enhance the student's holistic understanding of mathematical functions.

Power Function Characteristics

Power functions have unique characteristics that set them apart from other types of functions. The function \(y=x^{3/2}\) is a classic example of a power function, which essentially means that \(y\) is proportional to \(x\) raised to a fixed power, in this case \(3/2\). Such functions are known for their predictable shapes and behaviors based on the exponent's value.

For exponents greater than 1, the curve will exhibit a growth that accelerates as \(x\) increases—demonstrating an increasing rate of change or steepness in the graph. By contrast, power functions with exponents less than 1 (but greater than 0) display a gentler slope, increasing at a decelerating rate. In exercises and learning content, emphasizing the effect of the exponent on the graph's curvature provides students with a valuable shortcut to predicting the function's visual representation.

Graphing Functions

Graphing functions is a fundamental skill in visualizing mathematical concepts. When graphing the power function \(y=x^{3/2}\), it's vital to start by plotting points that are easy to calculate, such as (0,0), (1,1), and (4,8), as in the provided solution. By connecting these points smoothly and continuously, one outlines the curve's trajectory.

It's crucial that the curve reflects the properties inferred from our earlier discussion of the domain and range, specifically, the curve for \(y=x^{3/2}\) should never dip below the \(x\)-axis since there are no negative values in its range. Moreover, exercise improvement advice often includes verifying the curve against the behavior expected from the function's characteristics — in this case, a steadily steepening slope. Through attentive graphing, students can visually validate their understanding of domain, range, and the specific characteristics associated with the function at hand.