Analyze the functions and graphs provided in the image.
Understand the Problem
The image contains multiple mathematical functions and graphs. It seems to be a collection of examples or problems related to functions, including floor functions, piecewise functions, and their graphical representations.
Answer
$ y = \begin{cases} 0, & \text{if } x \in \mathbb{Z} \\ -1, & \text{if } x \notin \mathbb{Z} \end{cases} $
Answer for screen readers
$ y = \begin{cases} 0, & \text{if } x \in \mathbb{Z} \ -1, & \text{if } x \notin \mathbb{Z} \end{cases} $
Steps to Solve
- Identify the relevant equation
Based on the OCR output, the equation in question is $y = [x] + [-x]$, where $[x]$ represents the floor function.
- Understand the floor function
The floor function $[x]$ gives the greatest integer less than or equal to $x$. For example, $[3.2] = 3$, $[5] = 5$, $[-2.3] = -3$.
- Analyze the behavior when x is an integer
If $x$ is an integer, then $[x] = x$ and $[-x] = -x$. Therefore, $y = [x] + [-x] = x + (-x) = 0$
- Analyze the behavior when x is not an integer
If $x$ is not an integer, let $x = n + f$, where $n$ is an integer and $0 < f < 1$. Then $[x] = n$. Also, $-x = -n - f$, so $[-x] = -n - 1$. Thus, $y = [x] + [-x] = n + (-n - 1) = -1$.
- Summarize the result
$y = [x] + [-x] = \begin{cases} 0 & \text{if } x \in \mathbb{Z} \ -1 & \text{if } x \notin \mathbb{Z} \end{cases}$
$ y = \begin{cases} 0, & \text{if } x \in \mathbb{Z} \ -1, & \text{if } x \notin \mathbb{Z} \end{cases} $
More Information
The function $y=[x] + [-x]$ is an interesting function that returns 0 when $x$ is an integer and $-1$ when $x$ is not an integer.
Tips
One common mistake might be to assume that $[-x] = -[x]$ for all $x$. However, this is only true when $x$ is an integer. If $x$ is not an integer, $[-x] = -[x] - 1$. For instance, if $x = 3.5$, then $[x] = 3$, so $-[x] = -3$. But $[-x] = [-3.5] = -4$.
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