Problem 16 - Elementary and Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x-3 .\) Find each function and give its domain. $$ g+f $$

Step-by-Step Solution

Answer

The function \( g+f \) is \( 3x - 2 \) with domain \( (-\infty, \infty) \).

Step 1: Define the given functions

The given functions are stated as follows: \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \). We need to perform the operation \( g+f \), which means finding \( g(x) + f(x) \).

Step 2: Add the functions

To find \( (g+f)(x) \), we add the expressions for \( g(x) \) and \( f(x) \). \[ (g+f)(x) = g(x) + f(x) = (x - 3) + (2x + 1) \] Combine like terms to simplify the expression.

Step 3: Simplify the combined expression

Combine like terms from the previous step:\[ (g+f)(x) = x + 2x - 3 + 1 = 3x - 2 \] Thus, the function \( g+f \) is \( 3x - 2 \).

Step 4: Determine the domain of the combined function

Both \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \) are defined for all real numbers, as they are linear functions. Therefore, the domain of \( (g+f)(x) = 3x - 2 \) is also all real numbers, written in interval notation as \( (-\infty, \infty) \).

Key Concepts

Domain of a function

The domain of a function is a fundamental concept in algebra. It refers to all the possible input values (usually represented as \( x \)) that allow the function to return a valid output. For most functions in high school algebra, you look for undefined spots, like divisions by zero or negative square roots, to determine restrictions on the domain. However, with linear functions, which are often polynomials of the first degree, determining the domain is straightforward.

Linear functions, such as \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), have no such restrictions. This means their domain is all real numbers. In interval notation, this is expressed as \( (-\infty, \infty) \). In practice, this means you can substitute any real number into the equations, and both functions will be defined and produce consistent results. Understanding this helps you analyze various real-life situations modeled by such functions.

Linear functions

Linear functions are algebraic expressions that create straight lines when graphed on the coordinate plane. The general form of a linear function is \( f(x) = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept.

• The **slope** \( m \) indicates how steep the line is. A positive slope rises from left to right, whereas a negative slope falls.
• The **y-intercept** \( b \) is where the line crosses the y-axis, giving it a starting point when \( x = 0 \).

Understanding linear functions is crucial as they model many real-world scenarios, such as speed (distance over time) or costs (price per unit). In our example, \( f(x) = 2x + 1 \) has a slope of 2, suggesting it rises two units for every one unit it moves right. Similarly, \( g(x) = x - 3 \) features a slope of 1, indicating a gentler rise. Both functions' linearity explains their infinite domain from negative to positive infinity.

Combining functions in algebra

Combining functions includes operations such as addition, subtraction, multiplication, or division of functions. This method is crucial for constructing new functions based on existing ones, extending algebra's flexibility in modeling and simplifying problems.

When we say \( g + f \), we mean to add the outputs of both functions for every given input \( x \). Using the functions \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), adding them results in another function: \[ (g+f)(x) = g(x) + f(x) = (x - 3) + (2x + 1) = 3x - 2 \]Simplifying the combined function means collecting like terms, as shown above. By adding rates of change (slopes) and adjusting constants, we derived \( g + f \) as \( 3x - 2 \). This new function, being linear, assumes the same domain as the original functions, \( (-\infty, \infty) \). Understanding the process of combining functions helps develop deeper insights into how individual components impact a model's behavior.