In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = 2/(x+3), g(x) = 1/x

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation (fog)(x) represents the composition of f and g, meaning f(g(x)). To find this, substitute g(x) into f(x).
2. Step 2: Substitute g(x) = 1/x into f(x) = 2/(x+3). Replace every instance of 'x' in f(x) with g(x). This gives f(g(x)) = 2/((1/x) + 3).
3. Step 3: Simplify the expression for f(g(x)). Combine the terms in the denominator ((1/x) + 3) by finding a common denominator. This results in f(g(x)) = 2/(1/x + 3) = 2/((1 + 3x)/x).
4. Step 4: Further simplify f(g(x)) by dividing 2 by the fraction (1 + 3x)/x. This results in f(g(x)) = 2x/(1 + 3x).
5. Step 5: Determine the domain of f o g. The domain is the set of all x-values for which the composition is defined. For f(g(x)) = 2x/(1 + 3x), ensure that the denominator (1 + 3x) ≠ 0 and g(x) = 1/x is defined (x ≠ 0). Therefore, the domain excludes x = 0 and x = -1/3.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (f o g)(x) means applying g first and then applying f to the result. Understanding how to correctly perform this operation is essential for solving the problem.

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 composition of functions, the domain must consider the restrictions of both functions involved. Identifying these restrictions is crucial to determine the overall domain of the composed function.

Rational Functions

Rational functions are ratios of polynomials, and they often have restrictions based on the values that make the denominator zero. In this problem, both f(x) and g(x) are rational functions, so understanding how to identify and handle these restrictions is key to finding the correct domain for the composition.