Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$f(x)=\frac{4 x^{2}}{x^{2}+1}$$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{4 x^{2}}{x^{2}+1}\) is even and symmetric about the y-axis, but not symmetric about the origin.

Step 1: Check if the function is even

Substitute -x for x in the function \(f(x)=\frac{4 x^{2}}{x^{2}+1}\), yielding \(f(-x)=\frac{4 (-x)^{2}}{(-x)^{2}+1}=\frac{4 x^{2}}{x^{2}+1}\). Since this outcome is the same as the original function f(x), it is concluded that the function is even.

Step 2: Check if the function is odd

Check if the function is odd by comparing -f(x) to the result of substituting -x for x in the function. Since -f(x) is equal to \( -\frac{4 x^{2}}{x^{2}+1}\) and this formula is not equal to f(-x), the function is not odd.

Step 3: Conclude whether the function is even, odd, or neither

Since the function fulfills the criteria for being even and does not fulfill the criteria for being odd, it can be concluded that the function \(f(x)=\frac{4 x^{2}}{x^{2}+1}\) is even and symmetric about the y-axis, hence, not symmetric about the origin.

Key Concepts

Symmetry

Symmetry in functions helps us understand how a function behaves across different axes. It acts as a checkpoint that can reveal specific characteristics of the function, such as whether it is even or odd.
Check for the following types of symmetry:

• Y-axis symmetry: A function is symmetric about the y-axis if for every x, the function values at x and -x are identical, meaning that f(x) = f(-x). This is a characteristic of even functions.
• X-axis symmetry: A function is symmetric about the x-axis if for every value of y, y = -y. However, this is not common for functions, as it implies two values for y corresponding to one x, which fails the vertical line test.
• Origin symmetry: A function has symmetry about the origin if flipping it over both axes results in the same graph, i.e., f(-x) = -f(x). This characteristic is present in odd functions.

Recognizing symmetry can simplify the graphing process, verifying equations, and help in solving function-related problems.

Function Analysis

Analyzing functions often involves identifying key attributes such as domain, range, intercepts, and symmetries. Thorough function analysis allows you to predict the behavior of the function across its entire domain.
When examining functions for symmetry:

• Domain and Range: Ensure you understand what inputs (domain) your function can handle, and what outputs (range) actually occur.
• Intercepts: Determine where the graph of the function will intersect the x- or y-axis. These points can often illustrate the symmetry of the function.
• Symmetric points: For even functions, x-values mirror each other across the y-axis, which impacts the intercepts and overall graph.

In this function analysis, the examination starts with checking for evenness or oddness, which greatly influences how you approach graphing or solving equations related to the function.

Substitution Method

Substitution is a fundamental method utilized to easily analyze a function’s properties. It involves substituting one value into the function to obtain another, which can be essential for assessing symmetry.
The process typically involves:

• Substituting -x for x: To check if a function is even, substitute -x into the function. If f(x) = f(-x), the function is even.
• Substituting to find oddness: Substitute -x, if doing so yields f(-x) = -f(x), the function is odd.
• Verification: This method effectively verifies whether symmetries and unique characteristics associated with evenness or oddness are present in the function.

Applying the substitution method can reveal much needed insights on whether a function is symmetric, thereby aiding in problems of mathematical symmetry and visual graph analysis.