The domain is all possible x values for a function and the range is all possible y values for a function for which the function is defined.

For all functions that do not contain a square root, fraction, logarithm, or piecewise have a domain and range of all real numbers.

For example:

$$ 4x^4-5x^{29999999} $$

has a domain of all real numbers and a range of all real numbers because 4 and 29999999 are natural numbers and there are no fractions or irregularities seen.

Example 1: Square Root Function

$$ \text {Find the domain and range of}\space 2\sqrt{5-x^2}. $$

To solve a square root domain question, just set the inside of the square root greater than or equal to 0.

$$ 5-x^2\geq 0\to x^2\leq5\to x\leq\pm \sqrt{5}\to x\geq-\sqrt{5}\space,\space x\leq\sqrt{5} $$

To solve a square root range question, just think about how you obtained the domain again. This domain was obtained because the square root function can never be negative, so it can be concluded that the range is every value of y greater than or equal to 0. This info DOES NOT apply to cubic root functions because they can also have (-) x-values inside of them.

Example 2: Mixed Logarithm and Rational

$$ \text {Find the domain and range of }\frac{\log(x)}{x+5}. $$

To solve for the domain of a function with multiple restrictive functions, find the domain of each of the functions and then do the 'union' of those two intervals to get the final domain.

$$ \text {The domain of }\log(x)\space\text{is}\space x>0 $$

$$ \text {The domain of the rational function}\space \frac{1}{x+5} \space \text {is all real numbers except for} \space {x+5=0} $$

Thus, this can be solved and the domain becomes:

$$ (0,\infty)\cup(-\infty, \infty) = (0,\infty) $$

Since -5 is not in the given domain interval, we do not need to include it as part of our answer

To solve for the range of the function, simply think about what values are possible and what values are not. Since x=-5 is a vertical asymptote but is not in the domain, it does not need to be considered for the range. The logarithm has the only restriction that x has to be greater than or equal to 0, so this graph will have a full range unless there is a restriction due to the horizontal asymptote. Since this cannot be easily determined without a calculator, we can conclude that this function has a range of all real numbers for the purpose of Science Bowl.