Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=\sqrt{x^{2}+4} $$

Step-by-Step Solution

Answer

The function \( f(x) = \sqrt{x^{2} + 4} \) is even.

Step 1: Recall Definitions

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. It is odd if \( f(-x) = -f(x) \). If neither condition holds, the function is neither even nor odd.

Step 2: Determine \( f(-x) \)

Substitute \(-x\) into the function: \( f(-x) = \sqrt{((-x)^{2})+4} = \sqrt{x^{2} + 4} \). This shows that \( f(-x) = \sqrt{x^{2} + 4} \).

Step 3: Compare \( f(-x) \) and \( f(x) \)

Since \( f(-x) = \sqrt{x^{2} + 4} = f(x) \), the function satisfies the condition for being even.

Step 4: Conclusion

\( f(x) \) is even because \( f(-x) = f(x) \). Therefore, it does not satisfy the condition for being an odd function. Thus, \( f(x) \) is an even function.

Key Concepts

Function Properties

When we talk about functions in mathematics, we often refer to their properties, which help us understand their behavior and characteristics. A key property of functions that students encounter is the determination of whether a function is even, odd, or neither. An even function is defined as one that satisfies the condition \( f(-x) = f(x) \) for every \( x \) in its domain. This means that even functions have a symmetric shape around the y-axis. For a function to be odd, it must satisfy the condition \( f(-x) = -f(x) \), which results in rotational symmetry around the origin. If a function doesn't meet either of these conditions, it is classified as neither even nor odd.
Understanding these properties is important, as they help predict the graph's symmetry, simplifying the graphing process and solving equations.

Domain of a Function

The domain of a function is the complete set of possible values that the input, or x-values, can take. For the function \( f(x) = \sqrt{x^2 + 4} \), since we have a square root, the expression under the square root \( x^2 + 4 \) must be non-negative. However, in this case, \( x^2 + 4 \) is always positive for all real number values of \( x \) because \( x^2 \) is non-negative, and adding 4 ensures positivity. Therefore, the domain of \( f(x) \) is all real numbers, written as \( (-\infty, \infty) \).
The understanding of a function's domain is crucial as it determines the set of x-values you can substitute into the function, which influences the behavior and graph of the function as well.

Symmetry in Functions

Symmetry plays a pivotal role in understanding functions visually and algebraically. In the context of even and odd functions, symmetry can tell you a lot about the function's behavior. For an even function like \( f(x) = \sqrt{x^2 + 4} \), the symmetry is about the y-axis because \( f(x) = f(-x) \). This symmetry means that if you fold the graph along the y-axis, both sides would match. Such symmetry simplifies analysis and computations because it indicates predictable behavior on both sides of the y-axis.
Identifying symmetry helps in sketching the graphs and solving mathematical problems faster, as you can leverage symmetry to reduce calculations, make inferences about values, and understand function behavior more fully.