Problem 50 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Suppose \(f\) and \(g\) are both even functions. What can be said about \((f+g)(x)\) ? \((f g)(x) ?\) (b) Suppose \(f\) and \(g\) are both odd functions. What can be said about \((f+g)(x)\) ? \((f g)(x) ?\) (c) Suppose \(f\) is an even function and \(g\) is an odd function. What can be said about \((f+g)(x) ?(f g)(x) ?\)

Step-by-Step Solution

Answer

For two even functions \(f\) and \(g\), both the sum and product are even. For two odd functions, the sum is odd while the product is even. When \(f\) is an even function and \(g\) is an odd function, the sum is neither even nor odd but the product is odd.

Step 1: Sum and product of two even functions

Given that \(f\) and \(g\) are even functions. To verify the nature of \(f+g\) and \(fg\), try to express \((f+g)(-x)\) and \((fg)(-x)\). Using the properties of even functions, \((f+g)(-x) = f(-x) + g(-x) = f(x) + g(x)\), so \(f+g\) is an even function. Similarly, \((fg)(-x) = f(-x).g(-x) = f(x).g(x)\), so \(fg\) is also an even function.

Step 2: Sum and product of two odd functions

Given that \(f\) and \(g\) are odd functions. If expressed \((f+g)(-x)\) and \((fg)(-x)\) using the properties of odd functions, we get: \((f+g)(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x))\), hence \(f+g\) is an odd function. \((fg)(-x) = f(-x).g(-x) = -f(x).-g(x) = f(x).g(x)\), so \(fg\) is an even function.

Step 3: Sum and product of an even and an odd function

Given that \(f\) is an even function and \(g\) is an odd function. Now if we express \((f+g)(-x)\) and \((fg)(-x)\): \((f+g)(-x) = f(-x) + g(-x) = f(x) - g(x)\), so \(f+g\) is neither even nor odd. \((fg)(-x) = f(-x).g(-x) = f(x).-g(x) = -(f(x).g(x))\), so \(fg\) is an odd function.

Key Concepts

Sum of Functions

When discussing functions, the sum involves adding inputs from two given functions together. Let's see how this works for even and odd functions:

• **Even Functions:** These satisfy the condition where \(f(-x) = f(x)\). This symmetry around the y-axis has a special property, where if both functions are even, the sum \((f+g)(x)\) will also be even. This happens because the sum retains that y-axis symmetry: \(f(-x) + g(-x) = f(x) + g(x)\).


• **Odd Functions:** These have the property \(f(-x) = -f(x)\). When adding two odd functions, the sum \((f+g)(x)\) results in an odd function. This is because: \(f(-x) + g(-x) = -(f(x) + g(x))\), maintaining that symmetry in respect to the origin.


• **Mixed (Even and Odd):** When combining an even function and an odd function, the sum \((f+g)(x)\) becomes neither even nor odd, as the function does not have the same form either respecting the y-axis or origin symmetry.

Product of Functions

In mathematics, the product of functions involves multiplying their outputs together.

• **Even Functions:** When multiplying two even functions, \(f(x) \cdot g(x)\), the result is always an even function. This is because the product of two values that satisfy \(f(-x) = f(x)\) yields another even function with \((f \cdot g)(-x) = f(x) \cdot g(x)\).


• **Odd Functions:** Interestingly, when multiplying two odd functions, the result is an even function. Here's why: \((f \cdot g)(-x) = f(-x) \cdot g(-x) = (-f(x)) \cdot (-g(x)) = f(x) \cdot g(x)\), which follows the rule for even functions. Thus, \(f \cdot g(x)\) does not change sign.


• **Even and Odd Functions Together:** Multiplying an even and an odd function yields an odd function. Since: \(f(-x) \cdot g(-x) = f(x) \cdot (-g(x)) = -(f(x) \cdot g(x))\), \(f \cdot g(x)\) reflects the form of odd functions.

Function Properties

Knowing the properties of even and odd functions is crucial for predicting the behavior of their combinations.

• **Even Functions:** Characterized by symmetry around the y-axis, if you flip the graph along the y-axis, it looks the same. This property means that operations involving only even functions often preserve this symmetry.


• **Odd Functions:** Have symmetry about the origin, as if rotated 180 degrees. For these functions, their integrals over symmetric intervals are zero, which is an interesting application.


• **Mixing Even and Odd:** When combining these functions, it's essential to realize that symmetry might not be maintained unless other properties override this effect. In function composition, expect different results.

Understanding these core properties aids in grasping more complex problems and is essential for mastering calculus and algebra.