Question

If \(f(x)\) and \(g(x)\) are odd functions of \(x\), then which of the following is true? (a) \(f+g\) is odd, \(f g\) is odd (b) \(f+g\) is odd, \(f g\) is even (c) \(f+g\) is even, \(f g\) is odd (d) \(f+g\) is even, \(f g\) is even

Short answer

Answer: The sum of two odd functions is odd, and the product of two odd functions is even.

Step-by-step solution

Check the property of the sum of two odd functions

We want to determine if the function \(h(x) = f(x) + g(x)\) is also an odd function. To do this, we need to check if \(h(-x) = -(f(x) + g(x))\). Let's replace \(x\) with \(-x\): \(h(-x) = f(-x) + g(-x)\) We know that both \(f(x)\) and \(g(x)\) are odd, so we can replace \(f(-x)\) with \(-f(x)\) and \(g(-x)\) with \(-g(x)\), which results in: \(h(-x) = -f(x) - g(x)\) Because both functions are odd, we can conclude that their sum \(h(x) = f(x) + g(x)\) maintains the odd property.

Check the property of the product of two odd functions

Now let's examine the product of two odd functions to see if it's odd or even. We want to determine if the function \(p(x) = f(x) * g(x)\) is also odd. To do this, we need to check if \(p(-x) = -p(x)\). Let's replace \(x\) with \(-x\): \(p(-x) = f(-x) * g(-x)\) Again, given that \(f(x)\) and \(g(x)\) are odd functions, we can replace \(f(-x)\) with \(-f(x)\) and \(g(-x)\) with \(-g(x)\), which results in: \(p(-x) = (-f(x)) * (-g(x))\) Now, \((-a) * (-b) = a * b\), thus: \(p(-x) = f(x) * g(x)\), which is equivalent to \(p(x)\), not \(-p(x)\). Because the product of odd functions results in an even function, we can conclude that their product \(p(x) = f(x) * g(x)\) is even. Using the information from Steps 1 and 2, we can now answer the exercise: The sum of two odd functions is odd, and the product of two odd functions is even. So the correct answer is (b) \(f+g\) is odd, \(f g\) is even.

Key concepts

Sum of Functions

When dealing with functions, it's important to understand what happens when you add two functions together. The sum of functions, denoted as \( h(x) = f(x) + g(x) \), is simply adding the outputs of two functions for every input \( x \). Here, let's focus on adding odd functions.When both functions \( f(x) \) and \( g(x) \) are odd, we mean they have the property that for any \( x \), \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \).

• To determine if the sum \( h(x) = f(x) + g(x) \) is odd, we replace \( x \) with \( -x \) and get \( h(-x) = f(-x) + g(-x) \).
• By the odd function property, this is the same as \( -f(x) - g(x) = -(f(x) + g(x)) \).

This shows that the sum of two odd functions is indeed odd, as it satisfies the definition of an odd function: \( h(-x) = -h(x) \).

Product of Functions

The product of functions involves multiplying their outputs together for a given input \( x \). If we have two functions \( f(x) \) and \( g(x) \), the product is written as \( p(x) = f(x) \cdot g(x) \).For odd functions, the property is \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). To check the nature of the product:

• Substitute \( x \) for \( -x \) to find \( p(-x) = f(-x) \cdot g(-x) \).
• Since each function is odd, this translates to \((-f(x)) \cdot (-g(x))\).
• A negative times a negative is positive, so \( p(-x) = f(x) \cdot g(x) \).

This result is equivalent to \( p(x) \), indicating that \( p(x) \) is an even function because it satisfies \( p(-x) = p(x) \), not \( -p(x) \). The product of two odd functions results in an even function.

Function Properties

Understanding the properties of functions helps us predict behavior when they are combined. Functions have distinctive properties like being odd or even, which define how they react to transformations like negation of their inputs.

• An **odd function** satisfies \( f(-x) = -f(x) \). This means its graph is symmetric around the origin.
• An **even function** satisfies \( f(-x) = f(x) \). This signifies symmetry about the y-axis.

These properties play a crucial role when exploring operations like addition or multiplication of functions. Recognizing these properties can hint at whether resultant functions will be odd, even, or something else entirely.

Even Function

An even function is defined by its symmetry about the y-axis, which means it has the property \( f(-x) = f(x) \).Even functions exhibit a specific kind of symmetry:

• The y-values (outputs) remain the same when you swap the x-values (inputs) with their negatives.
• The graph doesn't change if you flip it over the y-axis.

This characteristic is the opposite of an odd function. While odd functions change signs with negation of their input, even functions remain unchanged.Common examples of even functions are quadratic functions like \( f(x) = x^2 \), where flipping the input leaves the output unaffected, showcasing the inherent natural symmetry. Understanding this symmetry is key to predicting how more complex expressions behave when summed or multiplied, especially when these properties intermingle as seen with the sum and product of odd functions.