Problem 27 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x \sqrt{1-x^{2}}$$

Step-by-Step Solution

Answer

The function \( f(x) = x \sqrt{1 - x^{2}} \) is odd.

Step 1: Identify the function

The function in question is \( f(x) = x \sqrt{1 - x^{2}} \). This function will be analyzed to find out whether it's odd, even, or neither.

Step 2: Test for evenness

To test for evenness, replace \( x \) with \( -x \) in the equation and simplify. If the new equation is identical to the original equation, then the function is even. The result becomes \( f(-x) = -x \sqrt{1 - (-x^{2})} = -x \sqrt{1 - x^{2}} \), which is not identical to the original function. Therefore, it is not even.

Step 3: Test for oddness

To verify if the function is odd, replace \( f(x) \) with \( -f(x) \) in the equation and simplify it. If the new equation is identical to the original equation, then the function is odd. In doing so, we get \( -f(x) = -x \sqrt{1 - x^{2}} \), which is identical to the original function, hence the function is odd.

Key Concepts

Even Functions

Even functions have a special kind of symmetry related to the y-axis of a graph. This means if you were to fold the graph along the y-axis, both sides would match perfectly. Visually, they are balanced on either side of the y-axis. To determine if a function is even, replace every instance of \(x\) with \(-x\) in the function and simplify it. If the resulting function is exactly the same as the original, then the function is even.

• An even function is mathematically represented as \(f(-x) = f(x)\).
• Examples include \(f(x) = x^2\) and \(f(x) = \cos(x)\).

In the given problem, substituting \(-x\) into \(f(x) = x \sqrt{1 - x^2}\) gives us \(-x \sqrt{1 - x^2}\), which is not the same as the original. Hence, this function is not even.

Odd Functions

Odd functions exhibit a different type of symmetry; they possess rotational symmetry around the origin. This means that if you rotate the graph 180 degrees about the origin, it would look the same. For a function to be classified as odd, the function \(-f(x)\) should equal the function \(f(-x)\).

• An odd function is defined by the equation \(f(-x) = -f(x)\).
• Common examples are \(f(x) = x^3\) and \(f(x) = \sin(x)\).

In our exercise, \(-f(x) = -x \sqrt{1 - x^2}\), which matches the previous calculation \(f(-x) = -x \sqrt{1 - x^2}\). Therefore, based on this analysis, the function \(f(x) = x \sqrt{1 - x^2}\) is classified as an odd function.

Function Symmetry

Function symmetry involves understanding how a graph reflects or rotates about certain points or axes, giving a function its characteristic shape and properties. Symmetry helps in predicting the behavior of functions and can reveal a lot about the function's structure without needing to calculate every value.

• Y-axis symmetry is indicative of even functions, where \(f(-x) = f(x)\).
• Origin symmetry is seen in odd functions, displaying \(f(-x) = -f(x)\).
• Functions that are neither even nor odd do not have these symmetries, appearing asymmetrical along both the y-axis and the origin.

Understanding symmetry not only makes it easier to graph functions but also provides insight into the function's behavior across different domains. In our example, the function \(f(x) = x \sqrt{1 - x^2}\) demonstrated symmetry around the origin, confirming its classification as an odd function.