Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the function is odd, even, or neither. \(g(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

The function \(g(x)=\sqrt{x}\) is neither even nor odd.

Step 1: Define Even Function

A function is considered even if for every input \(x\), \(f(x) = f(-x)\). This means the graph of the function is symmetric with respect to the y-axis.

Step 2: Define Odd Function

A function is considered odd if for every input \(x\), \(-f(x) = f(-x)\). This means the graph of the function is symmetric with respect to the origin.

Step 3: Evaluate \(g(x)\) and \(g(-x)\)

For the function \(g(x)=\sqrt{x}\), determine \(g(-x)\). Since square root of a negative number is not defined in real numbers, \(g(-x)\) is undefined.

Step 4: Analyze Even Function Condition

For \(g(x)\) to be even, \(g(x)\) must be equal to \(g(-x)\). However, as \(g(-x)\) is undefined, \(g(x)\) cannot be equal to \(g(-x)\). Hence, \(g(x)\) is not an even function.

Step 5: Analyze Odd Function Condition

For \(g(x)\) to be odd, \(-g(x)\) must be equal to \(g(-x)\). As \(g(-x)\) is undefined, this condition does not hold. Therefore, \(g(x)\) is not an odd function.

Step 6: Conclusion

Since \(g(x)\) is neither even nor odd, based on the symmetry criteria for even and odd functions, we conclude that \(g(x)=\sqrt{x}\) is neither even nor odd.

Key Concepts

Function Symmetry

Understanding function symmetry is key in determining if a function is even, odd, or neither. Symmetry in functions pertains to how the function's graph behaves in relation to specific lines or points.

Functions can exhibit the following types of symmetry:

• Even Symmetry: If a function is even, its graph is symmetric about the y-axis. This can be tested by checking if the equation \( f(x) = f(-x) \) holds true for all inputs \( x \) in the function's domain.
• Odd Symmetry: A function is odd if its graph is symmetric about the origin. This can be checked with the equation \( -f(x) = f(-x) \) for all \( x \) in the domain.

The concept of symmetry helps in predicting the behavior of the graph without having to plot every point. It also assists in simplifying various mathematical analyses and calculations.

Graphical Symmetry

Graphical symmetry refers to the visual attributes of a graph demonstrating evenness or oddness. Recognizing this symmetry can often make understanding a function's properties much easier.

Here's how you can visualize the symmetry:

• Y-axis Symmetry: If the graph looks identical on both sides of the y-axis, the function is even. Think of folding the graph along the y-axis. If both halves match, it shows even symmetry.
• Origin Symmetry: This means if you rotate the graph 180 degrees around the origin, it appears unchanged. Such symmetry indicates the function is odd.

While the graphical approach gives a clear visual confirmation, it's important to back it up with algebraic verification for accuracy, especially when evaluating more complex functions.

Function Evaluation

Function evaluation is the process of finding the output of a function for specific input values. It's an essential step when determining if a function is even, odd, or neither.

To evaluate the function \( g(x) = \sqrt{x} \):

• Finding \( g(-x) \): If you attempt to substitute \(-x\) into the function, since the square root of a negative number is not defined (in the real number system), \( g(-x) \) is undefined.
• This undefined result tells us the function cannot be even or odd based on traditional definitions applied to real numbers.

Through evaluation, you can directly see if the conditions for even or odd symmetry are unfulfilled, which happened in this example with \( g(x) \).

Mathematical Analysis

Mathematical analysis involves examining a function's characteristics methodically. For \( g(x) = \sqrt{x} \), analysis begins by applying definitions of symmetry to evaluate its nature.

Here's a breakdown of the approach:

• Even Function Test: An even function requires \( f(x) = f(-x) \). Since \( g(-x) \) is undefined, \( g(x) \) isn't even.
• Odd Function Test: For odd functions, the condition \( -f(x) = f(-x) \) must hold. Once more, undefined \( g(-x) \) leads us to conclude that it doesn't meet the criteria for odd functions.

The conclusion drawn from the analysis for \( g(x) = \sqrt{x} \) is that it is neither even nor odd. This methodical approach shows how definitions and function properties determine symmetry.