Problem 103 - College Algebra

Question

If a function is defined by an equation, explain how to find its domain.

Step-by-Step Solution

Answer

The domain of a function can be found by determining the type of the function and identifying restrictions that might limit the function's domain, such as negative values under square roots, zero value in denominators, or negative values in logarithmic functions. The domain is the set of all x-values that do not lead to these restrictions.

Step 1: Determine the given function

It's essential to identify the type of function given to you; this is because the methods of finding the domain depend on the kind of function. The function can be a linear, quadratic, square root, rational, logarithmic or a sin, cos and tan function, etc.

Step 2: Identify the possible restrictions

We need to pay attention to elements in the equation that might limit the domain. Restrictions can occur with the presence of square roots (the value under the square root must be greater than or equal to 0), rational fractions (the denominator cannot be zero), and logarithmic functions (the argument of the logarithmic function must be greater than zero). So we 'remove' x-values that would cause such problems.

Step 3: Clearly define the domain

After identifying and removing the restrictions, the remaining set of x values form the domain. The domain can be written in different forms: interval notation, inequality notation, or set notation. It's most common to write it in interval notation.