Common Functions: Videos & Practice Problems

Understanding common functions is crucial in mathematics. The constant function, defined as f(x)=c, has a domain of all real numbers and a range of the constant value c. The identity function f(x)=x has both domain and range as all real numbers. The square function f(x)=x² has a range from 0 to positive infinity. The cube function f(x)=x³ includes all real numbers for both domain and range. The square root function f(x)=x has a domain and range from 0 to positive infinity, while the cube root function f(x)=$x^3$ encompasses all real numbers.

Graphs of Common Functions

Understanding common functions is essential for mastering mathematical concepts, as these functions frequently appear in various contexts. Let's explore 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 (−∞, +∞), since any value of x can be input. However, the range is limited to the constant value itself, so in this case, the range is simply {2}.

Next, we have the identity function, expressed as f(x) = x. This function outputs the same value as the input, meaning if you input -1, the output is -1, and if you input 50, the output is 50. 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 square any real number. However, the range is restricted to non-negative values, starting from 0 to positive infinity, or [0, +∞), as the output of squaring a number cannot be negative.

In contrast, the cube function, given by f(x) = x³, includes all real numbers in both the domain and range. The graph extends infinitely in both the positive and negative directions, so both the domain and range are (−∞, +∞).

The square root function, represented as f(x) = √x, has more restrictions. The 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 output of the square root function is always non-negative.

Lastly, the cube root function, defined as f(x) = ∛x, allows for all real numbers in both the domain and range. This function can accept negative inputs and produce negative outputs, resulting in both the domain and range being (−∞, +∞).

Familiarity with these functions and their properties is crucial as they form the foundation for more complex mathematical concepts encountered in future studies.

Here’s what students ask on this topic:

What is the domain and range of the constant function?

The constant function is defined as $f\left(x\right)=c$, where $c$ is a constant. The domain of the constant function is all real numbers, which means you can input any value for $x$. This is represented as $(-\infty ,\infty )$. The range, however, is just the constant value $c$, meaning the output is always $c$ regardless of the input. So, the range is $\left\{c\right\}$.

What is the identity function and its domain and range?

The identity function is defined as $f\left(x\right)=x$. This means that whatever value you input for $x$, you get the same value as the output. The domain of the identity function is all real numbers, represented as $(-\infty ,\infty )$. Similarly, the range is also all real numbers, as the output can be any real number. So, the range is $(-\infty ,\infty )$.

What is the domain and range of the square function?

The square function is defined as $f\left(x\right)=x^2$. The domain of the square function is all real numbers, represented as $(-\infty ,\infty )$, because you can input any real number for $x$. The range, however, is from 0 to positive infinity, represented as $[0,\infty )$, because the output of $x^2$ is always non-negative.

What is the domain and range of the cube function?

The cube function is defined as $f\left(x\right)=x^3$. The domain of the cube function is all real numbers, represented as $(-\infty ,\infty )$, because you can input any real number for $x$. The range is also all real numbers, represented as $(-\infty ,\infty )$, because the output of $x^3$ can be any real number.

What is the domain and range of the square root function?

The square root function is defined as $f\left(x\right)=\sqrt{x}$. The domain of the square root function is from 0 to positive infinity, represented as $[0,\infty )$, because you cannot take the square root of a negative number. The range is also from 0 to positive infinity, represented as $[0,\infty )$, because the output of the square root function is always non-negative.

What is the domain and range of the cube root function?

The cube root function is defined as $f\left(x\right)=x^{\frac{1}{3}}$. The domain of the cube root function is all real numbers, represented as $(-\infty ,\infty )$, because you can take the cube root of any real number. The range is also all real numbers, represented as $(-\infty ,\infty )$, because the output of the cube root function can be any real number.