Problem 23 - Contemporary Precalculus

Question

Determine whether the function is even, odd, or neither . $$f(x)=e^{-x^{2}}$$

Step-by-Step Solution

Answer

Answer: The function \(f(x)=e^{-x^{2}}\) is even.

Step 1: Test for Evenness

To test if the function is even, we need to verify that \(f(x) = f(-x)\). We will compute \(f(-x)\) and then simplify to check if it is equal to \(f(x)\). $$ f(-x) = e^{-(-x)^{2}} = e^{-(x^{2})} $$ Since we find that \(f(x) = f(-x)\), the function is even.

Step 2: Test for Oddness

Even though we have already determined the function is even, we will still test for oddness for the sake of completeness and learning. To test for oddness, we need to verify if \(f(-x) = -f(x)\). From step 1, we know \(f(-x) = e^{-(x^{2})}\). Now, let's compute \(-f(x)\): $$ -f(x) = -e^{-x^{2}} $$ The given function does not satisfy the condition for an odd function, as \(f(-x) \ne -f(x)\).

Step 3: Conclusion

Since the function \(f(x)=e^{-x^{2}}\) satisfies the condition for an even function, it is an even function and not odd or neither.

Key Concepts

Symmetry in Functions

Understanding symmetry in mathematical functions is essential to analyzing their behavior and predicting certain properties about graphs of those functions. Symmetry in functions can commonly be described as even or odd.

A function is said to be even if its graph is symmetrical about the y-axis. This means if you were to fold the graph along the y-axis, both halves would align perfectly. A classic example of an even function is the cosine function, which exhibits this kind of symmetry.

Conversely, a function is considered odd if its graph is symmetrical about the origin. This translates to a 180-degree rotational symmetry. The sine function is a great example of an odd function with origin symmetry. Recognizing symmetry can thus help students predict the outcomes of function values and understand the behavioral patterns of various functions in calculus and other areas of mathematics.

In real-world applications, symmetry can make calculations and modelings easier, since knowing the nature of a function's symmetry can reduce the amount of work needed for certain integrations and transformations.

Testing for Evenness

The process of determining whether a function is even involves a simple test: checking if the value of the function at any positive input is the same as its value at the corresponding negative input. Mathematically, this is expressed as the condition that for every value of the variable, say 'x', the function satisfies the equation \( f(x) = f(-x) \).

Following these guidelines, let's consider the example of the function \( f(x)=e^{-x^{2}} \). By substituting '-x' into the function and simplifying, we find that \( f(-x) \) yields the same expression as \( f(x) \), therefore confirming that the function is even. In context, this implies the graph of \( f(x) \) would show y-axis symmetry. This property of a function being even can make certain analytical and computational tasks more manageable, like computing definite integrals over symmetric intervals.

Testing for Oddness

To determine if a function is odd, a different approach is used compared to testing for evenness. A function is odd if it satisfies the condition \( f(-x) = -f(x) \) for all values in its domain. The presence of the negative sign in this test reflects the fact that an odd function's graph has rotational symmetry about the origin.

Looking at our example function \( f(x)=e^{-x^{2}} \), we attempt to apply the oddness test. After computing, we see that \( f(-x) \) does not equal \( -f(x) \), which tells us that the symmetry required for a function to be odd is not present. It's important to note that a function can be neither even nor odd if it doesn't meet the criteria for either test, which means it has no symmetrical property with respect to the y-axis or the origin. Recognizing a function's lack of symmetry is just as crucial, as it may impact the approaches to solving equations and understanding function behavior.