In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the function \( \frac{f}{g} \) where \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \).
2. Step 2: Write the expression for \( \frac{f}{g} \). This is \( \frac{f(x)}{g(x)} = \frac{2x + 3}{x - 1} \).
3. Step 3: Determine the domain of \( \frac{f}{g} \). The domain of a rational function is all real numbers except where the denominator is zero.
4. Step 4: Set the denominator equal to zero and solve for \( x \). Solve \( x - 1 = 0 \) to find the value that is not in the domain.
5. Step 5: Conclude the domain. The domain of \( \frac{f}{g} \) is all real numbers except \( x = 1 \).

Key Concepts

Function Division

Function division involves creating a new function by dividing one function by another. In this case, f/g means taking the function f(x) = 2x + 3 and dividing it by g(x) = x - 1. The resulting function, f/g, can be expressed as (2x + 3) / (x - 1), which is essential for further analysis.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f/g, we must identify any values of x that would make the denominator g(x) = x - 1 equal to zero, as these values are excluded from the domain. In this case, x cannot equal 1, leading to the domain being all real numbers except x = 1.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. The function f/g is a rational function where the numerator is a linear polynomial (2x + 3) and the denominator is another linear polynomial (x - 1). Understanding the properties of rational functions, including their behavior near vertical asymptotes and discontinuities, is crucial for analyzing the function's characteristics.