Problem 58 - Calculus: One and Several Variables

Question

Given that \(f\) is defined for all real numbers, show that the function \(h(x)=f(x)-f(-x)\) is an odd function.

Step-by-Step Solution

Answer

\(h(x)\) satisfies the definition of an odd function, namely that \(h(-x) = -h(x)\). Thus, it's concluded that \(h(x) = f(x) - f(-x)\) is indeed an odd function.

Step 1: Definition of the function

We know that \(h(x) = f(x) - f(-x)\). This will serve as our starting point.

Step 2: Substitution of \(-x\) into \(h(x)\)

We want to find the behaviour of \(h\) when we substitute \(-x\) in place of \(x\). By doing this, we have \(h(-x) = f(-x) - f(-(-x)) = f(-x) - f(x)\). As can be seen, we now have the opposite function as before.

Step 3: Comparing \(h(-x)\) to \(-h(x)\)

Moreover, \(-h(x) = -(f(x) - f(-x)) = -f(x) + f(-x) = f(-x) - f(x) = h(-x)\). Thus, \(h(-x) = -h(x)\).

Key Concepts

Even Function

An even function is one where the function's graph is symmetrical around the y-axis. This means if you fold the graph along the y-axis, both sides would match perfectly. The mathematical identification of an even function is that for every input value \(x\), the value at \(x\) is equal to the value at \(-x\).

This relationship is expressed with the equation:

• \(f(x) = f(-x)\)

This means that whatever value or height the function attains at \(x\), it should also attain that same value at the negative \(x\).

For example, the function \(f(x) = x^2\) is even because:

• \(f(x) = x^2\) and \(f(-x) = (-x)^2 = x^2\)

Both expressions equal \(x^2\), thereby maintaining symmetry. This property is fundamental when examining a function to determine if it might be even or not.

Function Symmetry

Function symmetry refers to the property where a function behaves consistently in a mirrored fashion across a particular line or point. This concept includes both even and odd functions.

• Even functions, as said before, are symmetrical about the y-axis.
• Odd functions, on the other hand, exhibit symmetry about the origin.

In an odd function like \(h(x) = f(x) - f(-x)\), the graph will appear rotated 180 degrees about the origin. This means for any point \((x, y)\) on the graph, the point \((-x, -y)\) is also on the graph.

Mathematically, this can be shown by:

• \(h(-x) = -h(x)\)

This property is essential when attempting to classify a function as odd or even, as seen in the exercise where \(h(x)\) is shown to be odd by substituting \(-x\). Understanding symmetry helps us predict how the function behaves when its input is altered.

Function Properties

Understanding function properties is key in various areas of mathematics. These properties help identify special behaviors of functions and enable us to categorize them.

Key properties include:

• Symmetry: Even, odd, and periodic functions all share elements of symmetry in different forms.
• Continuity: This property describes if a function has no breaks, holes, or jumps in its graph. Every value within the domain should map to a distinct point on the function.
• Derivatives: These describe the rate at which a function changes at any point and are foundational in calculus.

Understanding whether a function is continuous or differentiable can have implications in physics, engineering, and more.

In the context of the exercise, comprehending that \(h(x) = f(x) - f(-x)\) is an expression of an odd function helps in analyzing its transformation effectively and predicting outcomes of substitutions. These aspects expose the broader significance and utility of function properties in solving real-world problems efficiently.