Find The Domain Of Composite Function

Finding the Domain of Composite Functions: A Comprehensive Guide

Determining the domain of a composite function is a crucial step in many mathematical and scientific applications. Understanding how to find this domain correctly ensures accurate calculations and meaningful interpretations of results. This comprehensive guide will walk you through the process, covering various scenarios and providing practical examples to solidify your understanding.

What is a Composite Function?

Before diving into the domain, let's briefly review composite functions. A composite function is a function formed by applying one function to the result of another. It's often represented as (f ∘ g)(x) or f(g(x)), meaning the function f is applied to the output of function g. The order matters; f(g(x)) is generally not the same as g(f(x)).

Understanding the 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. This means the function produces a real output (y-value) for every value in its domain. Values outside the domain result in undefined outputs – often division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

Common Restrictions:

• Division by Zero: The denominator of a fraction cannot be zero.
• Even Roots of Negative Numbers: You cannot take the even root (square root, fourth root, etc.) of a negative number in the real number system.
• Logarithms of Non-Positive Numbers: The argument of a logarithm must be positive.

Steps to Find the Domain of a Composite Function

Finding the domain of a composite function, f(g(x)), involves a two-step process:

1. Determine the Domain of the Inner Function (g(x)): First, find all possible values of x that make g(x) a valid input. This means identifying any restrictions based on the rules mentioned above (division by zero, even roots of negative numbers, logarithms of non-positive numbers).

2. Determine the Domain of the Outer Function (f(g(x))): Substitute g(x) into f(x). The domain of the composite function is now restricted by both the domain of g(x) and the permissible inputs of f(x) after the substitution. This often requires considering the range of g(x) and ensuring this range is within the acceptable input values for f(x).

In short: The domain of f(g(x)) is the set of all x-values such that:

• x is in the domain of g(x), and
• g(x) is in the domain of f(x).

Examples: Finding the Domain of Composite Functions

Let's illustrate these steps with various examples:

Example 1: Polynomial and Rational Functions

Let f(x) = x² + 1 and g(x) = x - 2. Find the domain of f(g(x)).

1. Domain of g(x): g(x) = x - 2 is a polynomial function; its domain is all real numbers, (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x - 2) = (x - 2)² + 1. This is also a polynomial, so its domain is all real numbers, (-∞, ∞).

Therefore, the domain of f(g(x)) is (-∞, ∞). Because both f(x) and g(x) have no domain restrictions, their composition inherits an unrestricted domain.

Example 2: Square Root and Polynomial Functions

Let f(x) = √x and g(x) = x - 4. Find the domain of f(g(x)).

1. Domain of g(x): g(x) = x - 4 is a polynomial, so its domain is (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x - 4) = √(x - 4). Here, the expression inside the square root must be non-negative: x - 4 ≥ 0, which means x ≥ 4.

Therefore, the domain of f(g(x)) is [4, ∞). The domain is limited by the requirement of a non-negative input for the square root function.

Example 3: Rational and Radical Functions

Let f(x) = 1/x and g(x) = √x. Find the domain of f(g(x)).

1. Domain of g(x): g(x) = √x. The domain is [0, ∞) because the input to the square root must be non-negative.

2. Domain of f(g(x)): f(g(x)) = f(√x) = 1/√x. Now, we have two restrictions: √x must be non-negative (as determined above) and √x cannot be zero because of division by zero. Combining these, we have x > 0.

Therefore, the domain of f(g(x)) is (0, ∞).

Example 4: Logarithmic and Polynomial Functions

Let f(x) = ln(x) and g(x) = x² + 1. Find the domain of f(g(x)).

1. Domain of g(x): g(x) = x² + 1 is a polynomial; its domain is (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x² + 1) = ln(x² + 1). Since x² is always non-negative, x² + 1 is always greater than or equal to 1. The natural logarithm is defined for positive inputs, and x² + 1 is always positive.

Therefore, the domain of f(g(x)) is (-∞, ∞).

Example 5: A More Complex Composite Function

Let's consider a more challenging example:

f(x) = √(x - 2) and g(x) = 1/(x-3). Find the domain of f(g(x)).

1. Domain of g(x): g(x) = 1/(x-3). The denominator cannot be zero, so x ≠ 3. The domain of g(x) is (-∞, 3) ∪ (3, ∞).

2. Domain of f(g(x)): f(g(x)) = √(1/(x - 3) - 2). The expression inside the square root must be non-negative: 1/(x - 3) - 2 ≥ 0. Solving for x:

   1/(x - 3) ≥ 2 1 ≥ 2(x - 3) 1 ≥ 2x - 6 7 ≥ 2x x ≤ 7/2

However, we also need to consider that the denominator x - 3 cannot be zero, so x ≠ 3. Also, remember the expression inside the square root has to be non-negative. Combining these conditions, we get (-∞, 3) ∪ (3, 7/2].

Therefore, the domain of f(g(x)) is (-∞, 3) ∪ (3, 7/2]

Conclusion

Finding the domain of a composite function requires a systematic approach. By carefully considering the domain restrictions of both the inner and outer functions, and how those restrictions interact, you can accurately determine the set of all possible input values for which the composite function is defined. This understanding is fundamental for accurate mathematical modeling and problem-solving across diverse fields. Remember to always check for potential issues like division by zero, negative values under even roots, and non-positive inputs for logarithms. Through practice and careful consideration of these potential issues, mastering the identification of composite function domains will become straightforward. Remember to always check your work! You can plug values from the edges of the proposed domain to confirm the function remains valid.