Domain and Range

The domain of a function is the complete set of possible values of the independent variable (usually x) for which the function is defined.

The range of a function is the complete set of possible values of the dependent variable (usually y) that results from using the function.

To find the domain, identify all the values of x that can be input into the function without causing any mathematical errors (like division by zero or taking the square root of a negative number).

To find the range, determine all the possible output values (y-values) that the function can produce based on its domain.

The function f(x) = 1/x has a limited domain because x cannot be 0.

The function f(x) = x^2 has a limited range of y ≥ 0, since squares of real numbers cannot be negative.

The domain is x ≥ 3, because the expression under the square root must be non-negative.

The range is y ≤ 0, since the function produces only non-positive values.

In a function that models the height of a ball thrown in the air, the domain could be the time (in seconds) the ball is in the air, and the range would be the height of the ball at those times.

It means the domain of the function includes every possible real number, often represented as (-∞, ∞).