How To Find The Domain Of Fog

How to Find the Domain of f(g(x)) (fog)

Finding the domain of a composite function, often denoted as f(g(x)) or fog, requires a systematic approach combining understanding of individual function domains and the impact of composition. This comprehensive guide will walk you through the process, equipping you with the knowledge to tackle various scenarios with confidence.

Understanding Domains of Functions

Before diving into composite functions, let's solidify our understanding of individual function domains. The domain of a function is the set of all possible input values (x-values) for which the function is defined. A function is considered undefined if:

• Division by zero: The function contains a denominator that can equal zero for certain x-values.
• Even roots of negative numbers: The function involves square roots, fourth roots, or other even roots of expressions that can be negative.
• Logarithms of non-positive numbers: The function contains logarithms (log or ln) of expressions that can be zero or negative.

Let's illustrate with examples:

Example 1: f(x) = 1/(x - 2)

The domain of f(x) is all real numbers except x = 2, since this would lead to division by zero. We can express this using interval notation as: (-∞, 2) U (2, ∞).

Example 2: g(x) = √(x + 3)

The domain of g(x) is all real numbers where x + 3 ≥ 0, meaning x ≥ -3. In interval notation: [-3, ∞).

Example 3: h(x) = ln(x)

The domain of h(x) is all positive real numbers (x > 0) because the natural logarithm is undefined for non-positive values. In interval notation: (0, ∞).

Finding the Domain of f(g(x))

The process of finding the domain of a composite function f(g(x)) involves two key steps:

1. Find the domain of the inner function, g(x). This sets an initial constraint on the possible input values.
2. Determine the values from the domain of g(x) that produce outputs (g(x)) that are within the domain of the outer function, f(x). This further refines the allowable input values to ensure that the entire composite function is defined.

Let's apply this to some examples:

Example 4: f(x) = 1/x and g(x) = x - 2

1. Domain of g(x): The domain of g(x) = x - 2 is all real numbers (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x - 2) = 1/(x - 2). This composite function is undefined when the denominator is zero, which happens when x = 2. Therefore, the domain of f(g(x)) is all real numbers except x = 2: (-∞, 2) U (2, ∞).

Example 5: f(x) = √x and g(x) = x + 3

1. Domain of g(x): The domain of g(x) = x + 3 is all real numbers (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x + 3) = √(x + 3). The square root is defined only when the expression inside is non-negative. Therefore, x + 3 ≥ 0, which means x ≥ -3. The domain of f(g(x)) is [-3, ∞).

Example 6: f(x) = ln(x) and g(x) = x²

1. Domain of g(x): The domain of g(x) = x² is all real numbers (-∞, ∞).

2. Domain of f(g(x)): f(g(x)) = f(x²) = ln(x²). The natural logarithm is defined only for positive arguments. Since x² is always non-negative, we need x² > 0. This inequality is true for all x except x = 0. Therefore, the domain of f(g(x)) is (-∞, 0) U (0, ∞).

Example 7: A more complex scenario

Let's consider a more challenging composite function:

f(x) = √(x - 1) and g(x) = 1/(x - 2)

1. Domain of g(x): The domain of g(x) = 1/(x - 2) is all real numbers except x = 2: (-∞, 2) U (2, ∞).

2. Domain of f(g(x)): f(g(x)) = √(1/(x - 2) - 1). For the square root to be defined, we need 1/(x - 2) - 1 ≥ 0. Let's solve this inequality:

   1/(x - 2) ≥ 1

   1/(x - 2) - 1 ≥ 0

   (1 - (x - 2))/(x - 2) ≥ 0

   (3 - x)/(x - 2) ≥ 0

   Now, we analyze the sign of this expression. The expression changes sign at x = 2 and x = 3. Testing intervals:

   • x < 2: (3 - x) is positive and (x - 2) is negative, so the expression is negative.
   • 2 < x < 3: (3 - x) is positive and (x - 2) is positive, so the expression is positive.
   • x > 3: (3 - x) is negative and (x - 2) is positive, so the expression is negative.

Therefore, the inequality (3 - x)/(x - 2) ≥ 0 is satisfied when 2 < x ≤ 3. Combining this with the domain restriction from g(x), the final domain of f(g(x)) is (2, 3].

Handling Different Function Types

The process remains consistent even when dealing with piecewise functions, trigonometric functions, or other types. Always remember to:

• Consider the domain restrictions of each individual function.
• Solve any inequalities that arise from roots, logarithms, or denominators.
• Combine the domain restrictions to find the overall domain of the composite function.

Practical Tips and Common Mistakes

• Work systematically: Break down the problem into smaller steps: find the domain of g(x), then consider the impact on the domain of f(x).
• Use interval notation: It provides a clear and concise way to represent the domain.
• Graphing can help: Visualizing the functions can offer insights into the domain of the composite function.
• Don't forget about the domain of the inner function: This is a crucial starting point.
• Carefully solve inequalities: Pay close attention to the signs when dealing with fractions and inequalities.

By following these steps and practicing with various examples, you'll master the skill of finding the domain of composite functions, a crucial concept in calculus and advanced mathematics. Remember, understanding function domains is fundamental for working with functions effectively and correctly interpreting mathematical models and real-world applications. The meticulous attention to detail required in determining the domain of a composite function underscores the importance of precision in mathematical reasoning. Mastering this skill enhances your overall mathematical proficiency and ability to work confidently with more complex mathematical structures.