Find f/g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the quotient of two functions, f(x) and g(x), which is represented as (f/g)(x) = f(x)/g(x). Additionally, you need to determine the domain of the resulting function.
2. Step 2: Write the given functions. f(x) = 2 + 1/x and g(x) = 1/x. Substitute these into the formula for (f/g)(x): (f/g)(x) = (2 + 1/x) / (1/x).
3. Step 3: Simplify the expression. To simplify (2 + 1/x) / (1/x), multiply the numerator (2 + 1/x) by the reciprocal of the denominator (x). This gives: (f/g)(x) = x(2 + 1/x).
4. Step 4: Distribute x in the simplified expression. Multiply x by each term in the parentheses: x(2) + x(1/x). This simplifies to 2x + 1.
5. Step 5: Determine the domain. The domain of a function is the set of all x-values for which the function is defined. For f(x) and g(x), x cannot be 0 because division by zero is undefined. Additionally, g(x) = 1/x cannot be zero, so x ≠ 0. Therefore, the domain of (f/g)(x) is all real numbers except x = 0.

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) and dividing it by g(x). This operation requires understanding how to manipulate algebraic expressions and the implications of division, particularly regarding the values that make the denominator zero.

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 rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Therefore, identifying the domain involves finding these critical points and excluding them from the set of possible inputs.

Asymptotes and Discontinuities

Asymptotes and discontinuities are important concepts in understanding the behavior of rational functions. Vertical asymptotes occur at values where the denominator is zero, indicating points of discontinuity in the function. Recognizing these points helps in determining the domain and understanding the graph's behavior near these critical values.