Problem 16 - Thomas Calculus

Question

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=(-x)^{3 / 2} $$

Step-by-Step Solution

Answer

The function is decreasing for \(-\infty < x < 0\) and has no symmetry.

Step 1: Analyze the Function

The given function is \( y = (-x)^{3/2} \). The term \((-x)\) indicates that the domain of \(x\) is restricted to negative values, since fractional exponents with an odd numerator and even denominator are only real for non-positive bases.

Step 2: Check for Symmetry

For symmetry with respect to the y-axis, the function must satisfy \( f(-x) = f(x) \). Here, \( y = (-x)^{3/2} \) becomes undefined or non-real for positive \(x\). Therefore, it is impossible to check for even function symmetry. Check for origin symmetry by substituting \( y = (-(-x))^{3/2} = x^{3/2} \) for positive \(x\): this doesn't equal \((-x)^{3/2}\). No symmetry is evident here.

Step 3: Determine Increasing/Decreasing Intervals

For \( y = (-x)^{3/2} \), differentiate to determine monotonicity. The derivative \( y' = \frac{3}{2}(-x)^{1/2} \cdot (-1) = -\frac{3}{2}(-x)^{1/2} \) shows that for negative \(x\), \(-x\) is positive, hence \( (-x)^{1/2} > 0 \). Therefore, \( y' < 0 \), indicating the function is decreasing for \(-\infty < x < 0\).

Step 4: Conclusion

The function \( y = (-x)^{3/2} \) does not exhibit symmetry and is strictly decreasing over its domain, \(-\infty < x < 0\). The function is undefined for non-negative \(x\).

Key Concepts

Function Symmetry

When analyzing function symmetry, we look for ways in which the graph of a function might mirror itself. There are two main types of symmetry to consider: symmetry with respect to the y-axis and symmetry with respect to the origin.
Symmetry with respect to the y-axis, or even symmetry, requires that the function satisfies the condition:

• \( f(-x) = f(x) \).

In the case of our function \( y = (-x)^{3/2} \), this symmetry cannot exist. Why? Because the function is undefined for positive \( x \) due to the negative base being raised to a fractional power, which usually involves complex numbers or becomes non-real for these values.
Now, for symmetry with respect to the origin, or odd symmetry, the requirement is:

• \( f(-x) = -f(x) \).

If you substitute to check origin symmetry, you'll find \( y = (-(-x))^{3/2} = x^{3/2} \) which does not equal \( -(-x)^{3/2}\). Hence, this function doesn't exhibit any symmetry. Understanding symmetry helps simplify problem-solving since symmetrical functions have predictable properties!

Increasing and Decreasing Intervals

Understanding intervals where functions increase or decrease helps in analyzing their behavior. For the function \( y = (-x)^{3/2} \), we check its derivative to determine this behavior.
The derivative reveals how the function's rate of change varies across its domain. If the derivative \( y' \) is positive over an interval, the function is increasing there; if it's negative, the function is decreasing. Let's find \( y' \):

• \( y' = -\frac{3}{2}(-x)^{1/2} \).

Here, \(-x\) remains positive for negative \(x\), making the expression \((-x)^{1/2} > 0\). Consequently, \( y' < 0 \), indicating a negative rate of change. This signifies a decreasing function. Thus:

• The function is strictly decreasing across its entire domain, \(-\infty < x < 0\).
• The function is undefined for non-negative \(x\), meaning it doesn't increase or decrease there; it just doesn't exist.

Grasping increasing and decreasing intervals helps you quickly identify where the function rises or falls.

Fractional Exponents

Fractional exponents represent roots and powers combined. For instance, in the expression \((-x)^{3/2}\), the fractional exponent \(3/2\) means:

• Raise \((-x)\) to the power of \(3\), producing \((-x)^3\).
• Take the square root, represented by the \(1/2\) part, of the result.

Fractional exponents can sometimes yield complex numbers, especially when dealing with negative bases and non-integer exponents. This is why the domain of \( y = (-x)^{3/2} \) is only negative real numbers, as positive \(x\) would lead to non-real results.
Understanding fractional exponents is essential for interpreting and predicting function behavior, especially in cases involving complex numbers or restricted domains. Appreciate them not just as numbers but as operations that guide how functions can be transformed and analyzed.