Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=x^{3 / 2}\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = x^{3 / 2}\) exists only in the first quadrant and the function is neither even nor odd.

Step 1: Graphing the function

The function \(f(x) = x^{3 / 2}\) is defined for all nonnegative values of x (since we can't take the square root of a negative number in the real number set). The graph starts at the point (0,0). As x increases, \(f(x)\) increases but at a decelerating rate. The resulting graph only exists in the first quadrant.

Step 2: Identifying the nature of the function

An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. Looking at the graph, it's clear that our function is not symmetric about the y-axis so it can't be even. It also isn't symmetric about the origin, so it can't be odd.

Step 3: Mathematically confirming the nature of the function

Mathematically, an even function is one for which \(f(-x) = f(x)\), while an odd function has \(f(-x) = -f(x)\). We can test these conditions for our function, but since \(f(-x)\) is not defined (sinde x must be nonnegative), we can confirm that the function is neither even nor odd.

Key Concepts

Function Graphing

Understanding how to graph functions is a crucial skill in mathematics. When graphing the function \(f(x) = x^{3/2}\), it's important to note that it is only defined for nonnegative \(x\) values. This is because we cannot take the square root of a negative number within the realm of real numbers. Therefore, when plotting this function, we start from the origin, point \((0,0)\). For positive values of \(x\), the function gradually increases, but at a decreasing rate. You will notice that the curve remains restricted to the first quadrant of the cartesian plane. Plotting this function involves observing how each x-value corresponds to an output when raised to the power of \(3/2\). As you graph \((x, f(x))\) points, the curve continuously rises without any reflective symmetry. This graphed behavior highlights a crucial aspect of function analysis: observing how outputs change with respect to their inputs, which aids in visualizing the function's behavior.

Symmetry in Functions

Symmetry in functions helps in identifying certain properties of functions quickly. For a function to be categorized as even, its graph must exhibit symmetry about the y-axis. This implies that for any input \(x\), the equation \(f(x) = f(-x)\) holds true. Conversely, odd functions display symmetry about the origin. This type of symmetry requires the condition \(f(x) = -f(-x)\) to be satisfied for all \(x\) in the domain.When we analyze the function \(f(x) = x^{3/2}\), neither of these symmetries are applicable. The function's only valid input range starts from \(x=0\) moving into positive x-values. Since \(-x\) is not a valid input for our function, we cannot fulfill the symmetry conditions necessary for a function to be either even or odd.Thus, by understanding these symmetry conditions and applying them correctly, we immediately ascertain that this function, \(f(x) = x^{3/2}\), is neither even nor odd.

Domain of a Function

The domain of a function is the set of all input values \(x\) for which the function is defined. For the function \(f(x) = x^{3/2}\), understanding its domain is key to plotting it correctly and analyzing its properties. Since the expression \(x^{3/2}\) includes a square root operation, the domain of this function is limited to nonnegative values. We can only take the square root of nonnegative numbers in the real number system.This means the domain of \(f(x)\) is

• \([0, \infty)\) - including zero and extending to positive infinity.

The limitation to nonnegative inputs directly affects the graph, which begins at the point \((0, 0)\) and extends towards positive values along the x-axis. Recognizing this domain constraint is vital when determining the function's graph, evaluating its symmetries, and understanding its overall behavior.By clearly defining the domain, students can better grasp why the function is neither even nor odd and how its graph shows a unidirectional increase.