A constant function assigns every real number to the same value. For example,
\begin{equation*}
f(x) = 4, \quad \text{$x$ any real number}
\end{equation*}
defines a constant function \(f\) that assigns every real number input to the value \(4\text{.}\) The graph of \(f\) is then the graph of the equation \(y = 4\) and consists of all points \((x,4)\text{,}\) where \(x\) is any real number. This is a horizontal line with slope zero.
Example6.12.The Identity Function.
The function \(i(x) = x\) is pretty boring. It identifies the input by returning the same value as its output. As such, we call it the identity function.
The symbol \(i\) is just the name I chose for the function in this example. For instance,
While it’s not a very interesting formula, it does play an important role in the algebra of functions later. As a linear function, its graph is a line passing through the origin with slope one.
The domain of \(i\) is \((-\infty,+\infty)\) and its range is \((-\infty,+\infty)\text{.}\)
Example6.13.Linear Functions.
A linear function is of the form \(L(x) = mx + b\text{,}\) where \(m\) determines the slope and \(b\) determines the \(y\)-intercept (and any \(x\)-intercept!). Use the sliders below to adjust \(m\) and \(b\) and observe their effect on the graph.
Figure6.14.Graph of \(y=mx+b\text{.}\)
Example6.15.The Reciprocal Function.
The reciprocal function returns the reciprocal of its input. It’s defined by \(R(x) = \frac{1}{x}\text{.}\)
Its domain is all real numbers \(x\neq 0\text{,}\) or in interval notation \((-\infty,0)\cup(0,+\infty)\text{.}\)
The reciprocal function outputs every non-zero real number, so that the range is \((-\infty,0)\cup(0,+\infty)\text{.}\) Observe three key features of its graph:
Near zero it is very large, because the reciprocal of a very small number is huge in magnitude. For instance \(R(0.001) = 1/(0.001) = 1000\text{.}\)
Far away from zero it is very small, because the reciprocal of a very large number is very small. For instance \(R(1000) = 1/(-1000) = 0.001\text{.}\)
The reciprocal of a negative number is also negative. So the graph of \(y = 1/x\) is below the \(x\)-axis on the interval \((-\infty,0)\text{.}\)
Example6.16.The Square Root Function.
The (principal) square root function is \(f(x) = \sqrt{x}\text{.}\) See Section 1.7.
However, \(f(-4)\) is undefined as the is no (real number) I can square to get \(-4\text{.}\) Thus, the domain of the square root function is \([0,+\infty)\text{.}\)
Checkpoint6.17.Cubing and Cube Root Functions.
The graphs of the cubing function \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) are plotted below. Identify the coordinates of three points on each graph by evaluating \(f\) and \(g\) and some convenient values. What is the domain and range of each function? Do you notice any relationship between the two graphs? How would you describe it?
