How do you determine the domain and range in interval notation?

Determining the domain and range of a function, with the results expressed in interval notation, involves the following steps:

To determine the domain: 1. Identify the function type: Whether the function is a polynomial, rational, radical, logarithmic, etc. 2. Find the values for which the function is defined: - For polynomials, the domain is all real numbers, because you can input any real number into a polynomial function. - For rational functions (fractions where the numerator or denominator is a polynomial), you must exclude values that make the denominator zero. - For radical functions (involving square roots, cube roots, etc.), ensure that the expression under the square root is non-negative (for real valued functions). - For logarithmic functions, the argument must be positive. 3. Write the domain in interval notation: - Use open interval (a, b) if a and/or b are not included in the domain (i.e., if the function is not defined at these points). - Use closed interval [a, b] if a and/or b are included (i.e., the function is defined and has a value at these points). - Use union (U) to join disjoint parts of the domain where the function is defined.

To determine the range: 1. Understand the function by examining its graph or applying calculus if you know how (finding the derivative can indicate the function's increasing and decreasing behavior). 2. Identify all possible values the function can output: - Look at the behavior of the function as the input goes to positive and negative infinity. - Consider special features of the graph such as asymptotes, plateaus, or bounds on the values of the function. 3. Write the range in interval notation much like the domain, using open or closed intervals and unions as appropriate.

Here's an example with a simple function: f(x) = sqrt(x - 2).

- The domain is all x such that x - 2 is greater than or equal to 0, so x is greater than or equal to 2. In interval notation, this is [2, ∞). - The range is all possible output values of f(x). Since f(x) is a square root function, it only outputs non-negative values and starts from the lowest value at f(2) = 0. So the range in interval notation is [0, ∞).