Problem 54 - Calculus: One and Several Variables

Question

State whether the function is odd, even, or neither. $$F(x)=x+\frac{1}{x}$$

Step-by-Step Solution

Answer

The function \(F(x)=x+\frac{1}{x}\) is odd.

Step 1: Test for evenness

To prove if the function is even, we need to prove that \(F(-x) = F(x)\). For \(F(x)=x+\frac{1}{x}\), if we replace \(x\) with \(-x\), we get \(F(-x)= -x - \frac{1}{x}\). As you see, \(F(-x) \neq F(x)\) so the function is not even.

Step 2: Test for oddness

To prove if the function is odd, we need to prove that \(F(-x) = -F(x)\). In our case: \(F(-x)=-x-\frac{1}{x}\), while \(-F(x) = -x - \frac{1}{x}\). Therefore, \(F(-x) = -F(x)\), hence the function is odd.

Key Concepts

Function Properties

Functions are fundamental in mathematics as they establish relationships between variable quantities. Understanding the properties of functions is essential for analyzing how they behave across different inputs. Two important properties, particularly in determining the symmetry of a function graph, are "even" and "odd" properties. These properties are particularly useful for graphing and understanding the geometric symmetry of functions.

Here are some essential properties of functions related to their symmetry:

• Symmetry: A function can have symmetry about the y-axis or origin, which is useful for graph interpretation.


• Intercepts: These are points where the function crosses the x-axis (x-intercepts) and y-axis (y-intercepts).


• Periodicity: This tells us if a function repeats its values in regular intervals. Common in sinusoidal functions.


• Continuity: This ensures a function is well-behaved, meaning there are no abrupt jumps or gaps.

By analyzing these properties, we gain insights into how functions behave and how we can predict their behavior in mathematical problems or real-world applications.

Even Function

An even function exhibits symmetrical behavior about the y-axis. This symmetry shows that if you reflect any point of the graph across the y-axis, you'll land on an equivalent point on the other side. This characteristic is defined mathematically by the equation
\[ f(-x) = f(x) \]
In practical terms, if you change the sign of the x-value (say by replacing \(x\) with \(-x\)), the result stays the same. Examples of even functions include common functions like \(f(x) = x^2\) or \(f(x) = \cos(x)\). Some features of even functions include:

• Graphs of even functions are symmetric which often simplifies graphing or solving equations.


• If a polynomial is even, all exponents of the variable are even numbers.


• These functions tend to appear frequently in physics when symmetry plays a role, such as in electric potentials.

Understanding even functions can simplify solving equations and predicting function behavior through visualization.

Odd Function

Odd functions are defined by their symmetry about the origin. This symmetry means that if you rotate the graph 180 degrees around the origin, the graph looks the same. Mathematically, this is expressed as:

\[ f(-x) = -f(x) \]
In essence, replacing \(x\) with \(-x\) in the function and multiplying the output by -1 should yield the original function's value, although reversed in sign. Common examples of odd functions include \(f(x) = x^3\) and \(f(x) = \sin(x)\). Here are some typical characteristics of odd functions:

• They often have terms that are all raised to an odd power if they are polynomial functions.


• Graphs of these functions will always pass through the origin if they are continuous.


• Odd functions are crucial in areas like mechanic systems where opposing forces come into play.

By understanding whether a function is odd or even, we significantly improve our ability to analyze and solve mathematical problems, especially in calculus and algebra.