Question

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

Short answer

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

Step-by-step solution

Evaluate the given function f(x)

The given function is: \(f(x) = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1}\)

Evaluate f(-x)

Substitute \(-x\) for \(x\) in the given function: \(f(-x) = \sqrt{(-x)^{2} -(- x) + 1} - \sqrt{(-x)^{2} + (-x) + 1} = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1}\)

Determine if the function is even

Compare \(f(x)\) and \(f(-x)\): \(f(x) = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1}\) \(f(-x) = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1}\) Since \(f(x) = f(-x)\), the function is even. The function \(f(x) = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1}\) is an even function.

Key concepts

Symmetry in Functions

Understanding symmetry in functions is foundational to recognizing patterns and behaviors within mathematical graphs. A function is said to be even if it exhibits symmetry about the y-axis. In simpler terms, for every point \( (x, y) \) on the function, the point \( (-x, y) \) will also be on the function. This property can be algebraically tested by replacing \( x \) with \( -x \) in the function's equation and simplifying. If the resulting expression is identical to the original function, it confirms the function is even.

For the exercise at hand, \( f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1} \), we applied this test and found that \( f(x) \) is indeed the same as \( f(-x) \) which establishes the function's evenness. Reflecting on the graph, you will observe the mirrored behavior across the y-axis, tying back to the concept of symmetry. This insight can greatly aid in graphing functions and understanding their inherent properties.

Function Transformation

Function transformation involves changing a function's position, shape, or orientation on a graph. Common transformations include shifting, reflecting, stretching, and compressing. Understanding how to manipulate a function's graph without altering its equation's core structure is a valuable skill in graph analysis and calculus.

In this context, if you were to manipulate the function \( f(x) \) from the exercise by changing the plus or minus signs within the radicals, or by multiplying by a factor, the graph's appearance would change accordingly. For instance, a reflection over the x-axis involves multiplying the entire function by -1. While transformations do not change the original function's even or odd nature, they provide a window into various function scenarios and how they respond to algebraic manipulation.

Radical Functions

Radical functions contain a variable within a root, typically a square root as seen in our example function. These functions often represent physical phenomena such as waves or acceleration in pendulum motion. The characteristics of radical functions can be quite complex due to their non-linear nature, which can cause them to behave differently from polynomial functions.

For example, with the function \( f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1} \), you are dealing with the square roots of binomials. The fact that it's an even function also indicates that, while the equation might look complicated, its graph will present a recognizable and symmetrical shape. By exploring radical functions in depth, students can develop a keen eye for these patterns, resulting in a more intuitive understanding of advanced mathematical concepts.