Problem 87 - Elementary Algebra

Question

Calculate \((f+f)(x)\) and state the restrictions to the domain. $$ f(x)=1 x $$

Step-by-Step Solution

Answer

\((f+f)(x) = 2x\) with domain all real numbers.

Step 1: Understanding the Function

The function given is \( f(x) = 1x \), which simplifies to \( f(x) = x \). This is a linear function, and linear functions typically have a domain of all real numbers. However, we need to evaluate any potential restrictions later.

Step 2: Calculating \((f+f)(x)\)

\( (f+f)(x) \) means we are adding the function \( f(x) \) to itself. So, we calculate: \[ (f+f)(x) = f(x) + f(x) = x + x = 2x \] Thus, \( (f+f)(x) = 2x \).

Step 3: Determining Domain Restrictions

Since we have determined that \( f(x) = x \) and \( (f+f)(x) = 2x \), both functions are linear. Linear functions, like \( 2x \), have a domain of all real numbers \((-fty, fty)\) and thus do not have any inherent domain restrictions. There are no denominators, square roots, or logarithms that could restrict the domain.

Key Concepts

Linear Function

When we talk about linear functions, we're referring to functions that can be graphically represented by a straight line. A linear function has the general form \( f(x) = ax + b \), where \( a \) and \( b \) are constants, and \( x \) represents the input. In the exercise given, \( f(x) = x \) is a linear function because it can be written in the form \( 1 \cdot x + 0 \), where the slope \( a = 1 \) and the y-intercept \( b = 0 \).
This means that for every unit increase in \( x \), \( f(x) \) increases by 1 unit too. This simplicity makes linear functions very predictable!
Linear functions are straightforward and typically have no restrictions on their domain, meaning they accept all real numbers as valid inputs.

Function Addition

Function addition is an interesting operation where two functions are combined by simply adding their outputs. If you have two functions, say \( f(x) \) and \( g(x) \), their sum is a new function defined as \( (f+g)(x) = f(x) + g(x) \). In this exercise, since we're asked to find \((f+f)(x)\), it means we're adding \( f(x) \) to itself.
This operation is quite simple for linear functions.
By replacing \( f(x) \) with \( x \), we get

• \((f+f)(x) = f(x) + f(x) = x + x = 2x\)

Function addition thus results in another linear function \( 2x \) and effectively scales the output by the sum of coefficients.

Domain Restrictions

To determine domain restrictions, we need to consider the elements in a function's formula that might limit what \( x \) can be. For instance, square roots, logarithms, or denominators (in fractions) can introduce restrictions because they can lead to non-real numbers or undefined expressions.
However, when dealing with linear functions like \( f(x) = x \) or \( (f+f)(x) = 2x \), things change. Linear functions are unrestricted unless additional factors are involved, which is not the case here.
This implies their domain is the set of all real numbers, written in interval notation as \((-fty, fty)\). This means no matter which real number you input, you will always get a valid, real number output. This property makes linear functions very versatile and easy to work with!