Tag: Step Functions

Draw the graphs of some step functions

by RoRi

For all $x \in \mathbb{R}$ define

$f(x) = [x], \qquad g(x) = [2x]$

where $[x]$ denotes the greatest integer less than or equal to $x$ . Draw the function $h(x)$ for each of the following on the interval $[-1,2]$ :

by RoRi

Define a function

$f(x) = \begin{cases} 1 & \text{for } 1 \leq x \leq 1; \\ 2 & \text{for } 1 < x \leq 2. \end{cases}$

(The function is not defined if $x < 0$ or $x > 2$ .)

Graph of $f$ .
Let $g(x) = f(2x)$ . Then the domain of $g$ is all $x$ such that $0 \leq x \leq 1$ . The graph is:
Let $h(x) = f(x-2)$ . Then the domain of $h$ is all $x$ such that $2 \leq x \leq 4$ . The graph is below.
Let $k(x) = f(2x) + f(x-2)$ . Then the domain is empty and so there is nothing to graph. (The domain is empty since $f(2x)$ is only defined for $0 \leq x \leq 1$ and $f(x-2)$ is only defined for $2 \leq x \leq 4$ . There are no $x$ that satisfy both of these conditions.)