Understanding common functions is essential for mastering mathematical concepts. This summary explores several key functions, their definitions, and their characteristics, including domain and range.

The constant function is defined as f(x) = c, where c is any constant number. For example, if f(x) = 2, the graph is a horizontal line at y = 2. The domain of a constant function is all real numbers, represented as (−∞, +∞), while the range is limited to the constant value, in this case, {2}.

Next, the identity function is expressed as f(x) = x. This function outputs the same value as the input, meaning if you input -1, the output is -1, and similarly for any other number. Both the domain and range of the identity function encompass all real numbers, (−∞, +∞).

The square function, defined as f(x) = x², produces a parabolic graph. The domain is all real numbers, (−∞, +∞), since you can input any real number. However, the range is restricted to non-negative values, [0, +∞), as the output cannot be negative.

In contrast, the cube function is given by f(x) = x³. This function includes all real numbers in both the domain and range, (−∞, +∞), allowing for both negative and positive outputs.

The square root function, represented as f(x) = √x, has more restrictions. Its domain is limited to non-negative values, [0, +∞), since you cannot take the square root of a negative number. The range is also [0, +∞), as the outputs are non-negative.

Lastly, the cube root function is defined as f(x) = ∛x. This function includes all real numbers in both the domain and range, (−∞, +∞), allowing for negative inputs and outputs.

These functions form the foundation for more complex mathematical concepts and are frequently encountered in various mathematical contexts. Familiarity with their properties, including domain and range, is crucial for success in future studies.