How to find the domain of a function

This section introduces the concept of finding the domain of a function, starting with polynomials. It defines polynomials and lists four conditions they must meet (no square roots of variables, no fractional variables, no negative powers on variables, no variables in the denominator). The domain for any polynomial is always all real numbers, which can be expressed in full sentences, with the symbol 'R', or in interval notation (-infinity, +infinity).

Several examples (Example 2 to Example 7) demonstrate how to determine the domain for various polynomial functions. These include quadratic functions (x^2 + 4x + 5), constant functions (like F(X) = 4, F(X) = 1/4, F(X) = -sqrt(3), F(X) = 0), and cubic functions (x^3 - 5). In all these cases, as they are polynomials, the domain is confirmed to be all real numbers.

This part explains how to find the domain of rational functions. Example 8 (2/(x+3)) shows that the denominator cannot be zero, leading to x ≠ -3. The domain is expressed as all real numbers except -3, or in interval notation (-infinity, -3) U (-3, +infinity). Example 9 (2/(x^2+3)) illustrates a case where the denominator x^2+3 is never zero for real numbers, so the domain is all real numbers.

This section covers finding the domain for functions involving square roots. Example 10 (sqrt(x+3)) demonstrates that the expression inside the square root must be greater than or equal to zero (x+3 ≥ 0), resulting in x ≥ -3. The domain is [-3, +infinity). Example 11 (sqrt(x^2+3)) shows that x^2+3 is always positive, so the domain is all real numbers. Example 12 (sqrt(x^2+2x-8)) involves factoring a quadratic expression to find the intervals where it is non-negative, leading to a domain of (-infinity, -4] U [2, +infinity).

This part focuses on logarithmic functions, explaining that the argument of a logarithm must be strictly positive (greater than zero). Example 13 (log(x+3)) shows that x+3 > 0, so x > -3, with the domain being (-3, +infinity). Example 14 (log(x^2+3)) demonstrates that x^2+3 is always positive, making the domain all real numbers. Example 15 (log(x^2+2x-8)) involves factoring and finding intervals where the expression is positive, resulting in a domain of (-infinity, -4) U (2, +infinity).

Example 16 (2^(x+3)) quickly establishes that the domain for exponential functions is all real numbers.

The final examples deal with more complex functions combining different types. Example 17 (sqrt(x+3)/(x-2)) requires both the numerator (square root argument) to be non-negative and the denominator to be non-zero. Example 18 ((x+3)/log(x-2)) considers the domain of the numerator (all real numbers), the argument of the logarithm (x-2 > 0), and the logarithm itself not equaling zero (log(x-2) ≠ 0), leading to x ≠ 3. Example 19 (sqrt(x+3)/(x^2-16)) combines a square root in the numerator and a denominator that cannot be zero (x^2-16 ≠ 0), resulting in x ≠ ±4.