Finding the domain of a function in Class 12 involves identifying all possible input values (x-values) for which the function is defined and produces a real number output. Essentially, you need to look for values of x that would cause the function to be undefined. Here's a breakdown of the common function types and how to determine their domains:
1. Understanding Domain
The domain of a function f(x) is the set of all real numbers 'x' for which f(x) is a real number. In simpler terms, it's all the x-values you can plug into the function without causing it to "break" (become undefined).
2. Common Restrictions and How to Handle Them
Certain function types have inherent restrictions that must be considered when finding the domain:
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Rational Functions (Fractions): The denominator cannot be zero.
- Method: Set the denominator equal to zero and solve for x. These x-values are excluded from the domain.
- Example: f(x) = 1/(x-2). The denominator (x-2) cannot be zero, so x ≠ 2. The domain is all real numbers except 2, which can be written as (-∞, 2) U (2, ∞).
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Radical Functions (Square Roots, etc.): The expression under an even-indexed radical (square root, fourth root, etc.) must be greater than or equal to zero.
- Method: Set the expression under the radical greater than or equal to zero and solve for x.
- Example: f(x) = √(x+3). We need x+3 ≥ 0, so x ≥ -3. The domain is [-3, ∞).
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Logarithmic Functions: The argument of a logarithm must be strictly greater than zero.
- Method: Set the argument of the logarithm greater than zero and solve for x.
- Example: f(x) = ln(x-1). We need x-1 > 0, so x > 1. The domain is (1, ∞).
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Trigonometric Functions:
- Sine and Cosine: Defined for all real numbers. Domain: (-∞, ∞).
- Tangent: Undefined where cosine is zero (x = π/2 + nπ, where n is an integer).
- Cotangent: Undefined where sine is zero (x = nπ, where n is an integer).
- Secant: Undefined where cosine is zero (x = π/2 + nπ, where n is an integer).
- Cosecant: Undefined where sine is zero (x = nπ, where n is an integer).
3. General Steps to Find the Domain
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Identify the type of function. Is it a rational function, radical function, logarithmic function, trigonometric function, or a combination?
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Identify any restrictions. Based on the function type, determine if there are any values of x that would make the function undefined (division by zero, negative under a square root, etc.).
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Solve for x. If there are restrictions, set up inequalities or equations to determine the values of x that must be excluded or included in the domain.
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Express the domain. Write the domain in interval notation, set notation, or using inequalities. Commonly, interval notation is used (e.g., (-∞, 5], (2, ∞), [-1, 3]).
4. Examples
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Example 1: f(x) = (x+1)/(x2 - 4)
- This is a rational function.
- Restriction: x2 - 4 ≠ 0
- Solving: x2 ≠ 4 => x ≠ ±2
- Domain: (-∞, -2) U (-2, 2) U (2, ∞)
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Example 2: f(x) = √(9 - x2)
- This is a radical function (square root).
- Restriction: 9 - x2 ≥ 0
- Solving: x2 ≤ 9 => -3 ≤ x ≤ 3
- Domain: [-3, 3]
5. Composite Functions
When dealing with composite functions (e.g., f(g(x))), you need to consider the domain of both the inner function g(x) and the outer function f(x). The domain of the composite function is the set of all x-values in the domain of g(x) such that g(x) is in the domain of f(x).
6. Expressing the Domain
The domain can be expressed in several ways:
- Interval Notation: Using parentheses ( ) for open intervals (excluding endpoints) and square brackets [ ] for closed intervals (including endpoints). Example: (-∞, 5]
- Set Notation: Using curly braces { } to list elements or define a set using a condition. Example: {x | x ≠ 2} (read as "the set of all x such that x is not equal to 2")
- Inequalities: Using inequalities to describe the range of values. Example: x > 3
Understanding these principles allows you to determine the domain of a wide variety of functions encountered in Class 12 mathematics.
