Determining the Domain of Mathematical Functions: A Guide with Examples

Domain

In mathematics, the domain refers to the set of all possible input values for a function or relation

In mathematics, the domain refers to the set of all possible input values for a function or relation. It represents the set of values for which a particular function is defined.

To determine the domain of a function, you need to consider any restrictions or limitations on the variables involved. Here are some guidelines to identify and find the domain of different functions:

1. Rational Functions:
A rational function is a function that is represented by the ratio of two polynomials. To find the domain of a rational function, you need to consider any values that would result in a denominator of zero since division by zero is undefined. So, you need to exclude any value that makes the denominator zero from the domain.

Example:

Consider the rational function f(x) = (x^2 – 9)/(x^2 – 4x + 3).
To find the domain, you need to determine the values of x that make the denominator (x^2 – 4x + 3) equal to zero:
x^2 – 4x + 3 = 0
Using factoring or the quadratic formula, you find that (x – 3)(x – 1) = 0, which means x = 3 or x = 1.

Therefore, the domain of f(x) is all real numbers except x = 3 and x = 1.

2. Radical Functions:
Radical functions involve square roots, cube roots, or nth roots. For these functions, you need to ensure that the radicand (the expression inside the radical) is non-negative since imaginary numbers might be involved for negative radicands.

Example:

Consider the square root function f(x) = √(4 – x^2).
To find the domain, you set the radicand greater than or equal to zero:
4 – x^2 ≥ 0
Using simple algebraic manipulation, you find that x^2 ≤ 4, which means -2 ≤ x ≤ 2.

Therefore, the domain of f(x) is -2 ≤ x ≤ 2.

3. Absolute Value Functions:
Absolute value functions involve the absolute value of a variable. These functions are defined for all real numbers, as the absolute value of a real number is always non-negative.

Example:

Consider the absolute value function f(x) = |x + 5|.
The domain of this function is all real numbers since it is defined for every value of x.

4. Polynomial Functions:
Polynomial functions are defined for all real numbers. So, the domain of a polynomial function is always the set of all real numbers.

Example:

Consider the polynomial function f(x) = x^3 + 2x^2 – 3x + 1.
The domain of this function is all real numbers.

These are just a few examples of finding the domain for different types of functions. Remember to analyze any specific restrictions or limitations on the variables in order to determine the correct domain.

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