Problem 3 - Intermediate Algebra

Question

Fill in the blanks. The _____ of the function \(f+g\) is the set of real numbers \(x\) that are in the domain of both \(f\) and \(g .\)

Step-by-Step Solution

Answer

The domain of \( f+g \) is the intersection of the domains of \( f \) and \( g \).

Step 1: Understanding the Function Sum

When we refer to the function \( f+g \), it means we are considering the sum of two functions \( f(x) \) and \( g(x) \) for the variable \( x \). To find \((f+g)(x)\), we need both \( f(x) \) and \( g(x) \) to be defined.

Step 2: Identifying the Domain

The domain of any function is the set of input values (\( x \)) for which the function is defined. For the function \( f+g \), \( x \) must be in the domains of both individual functions \( f \) and \( g \). This means the domain of \( f+g \) is constrained by the domains of both functions.

Step 3: Finding the Common Domain

To ensure \( f(x) + g(x) \) is defined, \( x \) must belong to both the domain of \( f \) and the domain of \( g \). Hence, the domain of \( f+g \) is the intersection of the domains of \( f \) and \( g \). This means \( x \) must be in both domains simultaneously to be valid for \( f+g \).

Key Concepts

Function Sum

When we talk about the function sum, we refer to adding two functions together, represented as \( (f+g)(x) \). Essentially, for every real number \( x \), you calculate both \( f(x) \) and \( g(x) \) separately and then add these values together to get the result of \( (f+g)(x) \). This allows us to combine the behaviors of \( f \) and \( g \) for each input \( x \).

The primary step to figure out the function sum is to confirm the definitions of both \( f(x) \) and \( g(x) \). If both functions are well-defined at a particular \( x \), you can proceed to calculate \( (f+g)(x) \) as \( f(x) + g(x) \). This provides a simple yet powerful way to merge functions and analyze their combined effect on their inputs.

Domain Intersection

The concept of domain intersection arises naturally when dealing with the function sum. The domain of a function is essentially the set of all possible inputs (\( x \) values) for which the function can give valid outputs. When adding two functions, \( f \) and \( g \), we require the inputs to be valid for both \( f \) and \( g \).

Mathematically, this means that the domain of \( (f+g) \) is the intersection of the domains of \( f \) and \( g \). For instance, if \( f(x) \) is defined for \( x > 0 \) and \( g(x) \) is defined for \( x < 5 \), then the function \( (f+g)(x) \) can only be defined for values where both conditions are satisfied. Therefore, the domain intersection would be \( 0 < x < 5 \), where both functions can operate correctly.

Key points to remember include:

• The domain intersection ensures \( (f+g)(x) \) remains well-defined by restricting \( x \) to where both \( f \) and \( g \) overlap.
• This concept helps prevent undefined operations that might occur if one function isn't defined at a certain value.

Real Numbers

Real numbers form the backbone of most domains for functions we encounter. In the realm of functions, when we talk about the domain, we typically mean the set of real numbers over which the function is defined and working correctly. Real numbers are any possible number that can be found on the number line, including integers, fractions, and irrational numbers like \( \sqrt{2} \) or \( \pi \).

Functions often find their inputs within this continuum of real numbers. When analyzing the domain of a function like \( (f+g) \), understanding that it includes only real numbers where both \( f \) and \( g \) are defined becomes crucial. Real numbers allow functions to behave predictably, such as linear continuity and differentiability, leading to consistent results when adding functions together.

Some important reminders about real numbers:

• Real numbers include negatives, positives, zero, rationals (like 1/2, 3/4), and irrationals (like \( \sqrt{2} \), \( \pi \)).
• They're critical in determining where a function is defined and valid, which affects calculations like the sum of two functions.