Graphing Common Functions: Videos & Practice Problems

Understanding common functions like constant, identity, square, cube, square root, and cube root is essential in algebra. The constant function $f\left(x\right)=c$ has a domain of all real numbers and a range of the constant $c$. The identity function $f\left(x\right)=x$ includes all real numbers in both domain and range. The square function $f\left(x\right)=x^2$ has a domain of all real numbers but a range from 0 to infinity. The cube and cube root functions cover all real numbers in domain and range, while the square root function restricts domain and range to non-negative values. These functions illustrate key concepts like domain, range, and the behavior of polynomial and radical expressions.

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, 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 for the identity function are 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 is given by f(x) = x³. This function includes all real numbers in both the domain and range, (−∞, +∞), as cubing any real number yields a real number, whether positive or negative.

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

In summary, familiarity with these common functions—constant, identity, square, cube, square root, and cube root—along with their respective domains and ranges, is crucial for further studies in mathematics. Understanding these concepts will aid in graphing and analyzing functions effectively.

Here’s what students ask on this topic:

What is the domain and range of the constant function $f\left(x\right)=c$?

The constant function is defined as $f\left(x\right)=c$, where $c$ is any constant number. The domain of this function is all real numbers, which means you can input any value for $x$ from negative infinity to positive infinity. This is because the output does not depend on $x$. The range, however, is limited to the single value $c$, since the function always outputs the constant $c$ regardless of the input. So, the domain is $(\infty ,\infty )$ and the range is $\{c\}$.

How do the domain and range of the square function $f\left(x\right)=x^2$ differ from those of the cube function $f\left(x\right)=x^3$?

The square function $f\left(x\right)=x^2$ has a domain of all real numbers, meaning any real number can be input. However, its range is limited to $\left[0,\infty \right)$ because squaring any real number results in a non-negative output; the graph forms a parabola opening upwards. In contrast, the cube function $f\left(x\right)=x^3$ also has a domain of all real numbers, but its range is all real numbers as well, $(\infty ,\infty )$. This is because cubing preserves the sign of $x$, allowing outputs to be negative or positive. The cube function's graph passes through the origin and extends infinitely in both directions.

Why is the domain of the square root function $f\left(x\right)=\sqrt{x}$ restricted to non-negative values?

The square root function $f\left(x\right)=\sqrt{x}$ is restricted to non-negative values for its domain because the square root of a negative number is not defined within the real numbers. This means you cannot input negative values for $x$ if you want a real output. Therefore, the domain is $\left[0,\infty \right)$. Correspondingly, the range is also $\left[0,\infty \right)$ because the square root function outputs only non-negative values. The graph starts at the origin and increases to the right, never extending into negative $y$ values.

What are the domain and range of the cube root function $f\left(x\right)=\sqrt{x}{3}^{}$?

The cube root function $f\left(x\right)=\sqrt{x}{3}^{}$ (more precisely, $f\left(x\right)=\sqrt{x}$ with cube root notation) has a domain and range of all real numbers. Unlike the square root function, the cube root can accept negative inputs because the cube root of a negative number is a negative number. This means the domain is $(\infty ,\infty )$. Similarly, the range is also all real numbers since the output can be any real number, positive or negative. The graph passes through the origin and extends infinitely in all directions, reflecting the symmetry of the cube root function.

How does the identity function $f\left(x\right)=x$ illustrate the concept of domain and range?

The identity function $f\left(x\right)=x$ is a simple yet powerful example illustrating domain and range. Its domain is all real numbers because you can input any real number $x$. The output is exactly the same as the input, so the range is also all real numbers. This means the function maps every real number to itself, making the graph a straight line passing through the origin with a slope of 1. This function helps students understand that domain refers to all possible inputs, while range refers to all possible outputs.