How To Find The Domain Of F O G

How to Find the Domain of f o g: A Comprehensive Guide

Finding the domain of a composite function, specifically f o g (often written as (f ∘ g)(x)), requires a methodical approach. It's not simply a matter of combining the domains of f(x) and g(x). Instead, we need to consider the interplay between the two functions. This guide will walk you through the process step-by-step, providing examples and addressing common pitfalls.

Understanding Composite Functions

Before diving into the domain, let's solidify our understanding of composite functions. A composite function is a function within a function. In the notation f o g, we first apply the function g(x) to the input x, and then we use the output of g(x) as the input for the function f(x). This can be expressed as:

(f o g)(x) = f(g(x))

This means we substitute the expression for g(x) wherever we see 'x' in the expression for f(x).

Determining the Domain of f o g: A Step-by-Step Process

The process of finding the domain of f o g involves three crucial steps:

1. Find the Domain of g(x): This is the initial step. The domain of g(x) represents all possible input values for g(x) that produce a real output. We need to identify any restrictions, such as division by zero, even roots of negative numbers, or logarithms of non-positive numbers.

2. Find the Range of g(x): The range of g(x) is the set of all possible output values of g(x). This is crucial because the output of g(x) becomes the input for f(x).

3. Find the Domain of f(x) considering the Range of g(x): This is where things get interesting. We aren't simply looking at the domain of f(x) in isolation. Instead, we're focusing on the values within the range of g(x) that are acceptable inputs for f(x). Any value in the range of g(x) that is not in the domain of f(x) must be excluded from the domain of (f o g)(x).

Example 1: Polynomial and Polynomial

Let's start with a relatively straightforward example.

g(x) = x + 2

f(x) = x²

Step 1: Domain of g(x)

The domain of g(x) is all real numbers, denoted as (-∞, ∞). There are no restrictions on the input values for a simple linear function.

Step 2: Range of g(x)

The range of g(x) is also all real numbers, (-∞, ∞). A linear function with a non-zero slope will cover the entire real number line.

Step 3: Domain of f(x) considering the Range of g(x)

The function f(x) = x² also accepts all real numbers as inputs. Since the range of g(x) is all real numbers, and all real numbers are acceptable inputs for f(x), the domain of (f o g)(x) is all real numbers, (-∞, ∞).

(f o g)(x) = f(g(x)) = f(x+2) = (x+2)²

Example 2: Rational Function and Polynomial

Now let's increase the complexity.

g(x) = x - 3

f(x) = 1/(x + 1)

Step 1: Domain of g(x)

The domain of g(x) is all real numbers, (-∞, ∞).

Step 2: Range of g(x)

The range of g(x) is also all real numbers, (-∞, ∞).

Step 3: Domain of f(x) considering the Range of g(x)

The domain of f(x) is all real numbers except x = -1, because this leads to division by zero. However, the range of g(x) includes -1. This means that there's a value within the range of g(x) that is not in the domain of f(x). Therefore, we must exclude the value of x that makes g(x) = -1. Solving x - 3 = -1 gives x = 2.

The domain of (f o g)(x) is all real numbers except x = 2, written as (-∞, 2) U (2, ∞).

(f o g)(x) = f(g(x)) = f(x - 3) = 1/((x - 3) + 1) = 1/(x - 2)

Example 3: Square Root Function and Quadratic Function

This example introduces a square root, adding another layer of complexity.

g(x) = x²

f(x) = √(x - 4)

Step 1: Domain of g(x)

The domain of g(x) is all real numbers, (-∞, ∞).

Step 2: Range of g(x)

The range of g(x) is [0, ∞). The square of any real number is always non-negative.

Step 3: Domain of f(x) considering the Range of g(x)

The domain of f(x) is [4, ∞), because the expression inside the square root must be non-negative. The range of g(x) is [0, ∞), but only the values from 4 onwards are acceptable inputs for f(x). Therefore, we need to find the values of x for which g(x) ≥ 4.

This means x² ≥ 4, which implies x ≥ 2 or x ≤ -2.

Therefore, the domain of (f o g)(x) is (-∞, -2] U [2, ∞).

(f o g)(x) = f(g(x)) = f(x²) = √(x² - 4)

Example 4: Logarithmic Function and Rational Function

This example involves logarithmic functions, requiring careful consideration of the argument's restrictions.

g(x) = 1/(x-2)

f(x) = ln(x)

Step 1: Domain of g(x)

The domain of g(x) is all real numbers except x = 2, (-∞, 2) U (2, ∞).

Step 2: Range of g(x)

The range of g(x) is (-∞, 0) U (0, ∞).

Step 3: Domain of f(x) considering the Range of g(x)

The domain of f(x) = ln(x) is (0, ∞). Since the range of g(x) includes negative numbers and zero, which are not allowed for the natural logarithm, we need to identify the values of x for which 1/(x-2) > 0.

This inequality is satisfied when x > 2. Therefore, the domain of (f o g)(x) is (2, ∞).

(f o g)(x) = f(g(x)) = f(1/(x-2)) = ln(1/(x-2))

Common Mistakes to Avoid

• Ignoring the range of g(x): The most common mistake is to simply consider the domains of f(x) and g(x) individually and combine them. The range of g(x) is crucial.
• Incorrectly solving inequalities: When dealing with square roots, logarithms, or other functions with restricted domains, carefully solving inequalities is essential.
• Not considering all restrictions: Make sure to account for all potential restrictions, such as division by zero, negative arguments in square roots, or zero or negative arguments in logarithms.

Conclusion

Finding the domain of a composite function requires a systematic approach that carefully considers both the domain of each function and the interplay between them. By following the three-step process detailed above and carefully examining the range of the inner function, you can accurately determine the domain of f o g, even for complex composite functions. Remember to always check your work and be meticulous in considering all possible restrictions. This will enhance your understanding of functions and their properties, crucial for success in advanced mathematical studies.