Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\sqrt{x^{2}+1}$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Define even, odd, and neither functions

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

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

Given \(f(x) = \sqrt{x^2 + 1}\), we need to evaluate \(f(-x)\). Substitute \(-x\) into the function: \[ f(-x) = \sqrt{(-x)^2 + 1} \].

Step 3: Simplify \(f(-x)\)

Simplify the expression \(\sqrt{(-x)^2 + 1}\). Since \((-x)^2 = x^2\), this becomes \(\sqrt{x^2 + 1}\). Therefore, \(f(-x) = \sqrt{x^2 + 1}\).

Step 4: Compare \(f(-x)\) to \(f(x)\)

Since \(f(-x) = \sqrt{x^2 + 1}\) and \(f(x) = \sqrt{x^2 + 1}\), we find that \(f(-x) = f(x)\).

Step 5: Determine if the function is even, odd, or neither

Since \(f(-x) = f(x)\), the function is even. It satisfies the condition for even functions and does not satisfy the condition for odd functions.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine the corresponding output. This process helps understand how the function behaves and is crucial in identifying the nature of the function as either even or odd, among other properties. For example, in the equation provided, we had to evaluate the function at

• \(f(x) = \sqrt{x^2 + 1}\)
• we needed to determine \(f(-x)\).

This involves replacing \(x\) with \(-x\) in the expression:\[f(-x) = \sqrt{(-x)^2 + 1}\]Calculating this and simplifying helps us understand how the function responds to inputs, including whether it might be symmetric with respect to particular operations like negation. Knowing how to evaluate a function by substitution is a foundational skill in calculus and algebra, providing insight into the function's overall behavior.

Function Properties

Every function has specific properties that define how it behaves in relation to various operations. Two important properties to examine are whether a function is even or odd. An **even function** satisfies the equation \(f(-x) = f(x)\). Geometrically, such functions are symmetrical about the y-axis.

• In the original exercise, since \(f(-x) = \sqrt{x^2 + 1} = f(x)\), the function is even.

An **odd function** would satisfy \(f(-x) = -f(x)\) and exhibit symmetry about the origin. Functions that do not fit either condition are neither even nor odd. Understanding whether a function is even, odd, or neither helps in analyzing the function's graph and predicting its behavior under transformations.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This makes evaluation easier and reveals properties of the function more clearly. In the given exercise, simplifying involved recognizing that squaring any number, including negative ones, results in a positive value:

• \((-x)^2 = x^2\)
• thereby making \(\sqrt{(-x)^2 + 1} = \sqrt{x^2 + 1}\).

Such simplification is key to verifying the function's properties, like in determining it is even. Simplification in algebra largely follows set rules like those involving exponents and radicals, enabling clearer interpretations and easier calculations. It is a foundational skill in mathematics that helps unravel complex expressions efficiently.