Problem 33 - Calculus

Question

Are the functions even, odd, or neither? $$f(x)=e^{x^{2}-1}$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Understand Function Symmetry Tests

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

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

Compute \(f(-x)\) for the given function:\[f(-x) = e^{(-x)^{2}-1} = e^{x^{2}-1}\]

Step 3: Check for Evenness

Compare \(f(x)\) with \(f(-x)\):Since \(f(-x) = e^{x^{2}-1} = f(x)\), the function satisfies \(f(x) = f(-x)\). This means the function is even.

Step 4: Verify Non-Oddness

For completeness, confirm \(f(x)\) is not odd:We have \(f(-x) = e^{x^{2}-1}\), but \(-f(x) = -e^{x^{2}-1}\). Clearly, \(f(-x) eq -f(x)\).

Key Concepts

Function Symmetry

Function symmetry is an important concept that helps to understand and categorize functions based on how their graphs look. Functions can be symmetric with respect to various transformations or the origin:

• **Even Functions:** These functions satisfy the condition that the function value at a positive input is the same as at its negative counterpart. Mathematically, this is expressed as \(f(x) = f(-x)\) for all \(x\) in the domain of the function.
  
• **Odd Functions:** For odd functions, the sum of the function value at a point and its inverse is zero. This is formalized as \(f(-x) = -f(x)\).
  

These conditions provide a structured way to determine the type of symmetry a function may have.

Applying this to the given function \(f(x) = e^{x^{2}-1}\), when you calculate \(f(-x)\), notice it simplifies to \(f(-x) = e^{(-x)^{2}-1} = e^{x^{2}-1} = f(x)\). This demonstrates that the given function satisfies the even function condition, making it symmetric about the y-axis.

Exponential Functions

Exponential functions are a crucial class of functions in mathematics, represented by formulas where the variable is an exponent. These functions have key characteristics which make them unique:

• The general form is \(f(x) = a^x\), where \(a\) is a positive constant.
  
• They are continuous for all real numbers \(x\) and have a base greater than zero.
  

For the specific function in this exercise, \(f(x) = e^{x^{2}-1}\), the function includes the natural exponential base \(e\), which is approximately 2.718.
The expression in the exponent, \(x^2 - 1\), alters our typical exponential function by adding a quadratic element. This affects how the function behaves, particularly in terms of growth rate and symmetry, as seen in the evenness of the function.

Calculus Problem Solving

Solving calculus problems effectively requires a clear understanding of different mathematical properties and techniques.

When tackling problems related to function symmetry, for example, the main focus is to determine the characteristics like evenness or oddness. This exercise shows one such approach:

• Start by substituting \(-x\) into the function to find \(f(-x)\).
  
• Compare \(f(-x)\) with the original function \(f(x)\). Check whether they are equal to conclude if the function is even.
  
• Check if the negative of the original function, \(-f(x)\), equals \(f(-x)\) to determine if it’s odd.
  

This systematic approach not only aids in classifying the function correctly but also deepens one's understanding of how these different types of functions behave.