Question

Which graph represents the step function f(x)=⌊x−1⌋ ?

Answer 1

Answer: The correct option is fourth.

Step-by-step explanation:

The given function is a greatest integer function or a step function.

`f(x)=\left \lfloor x-1 \right \rfloor`

In these type of functions the value of the function is always an integer and it is a piecewise function which is defined as,

`\left \lfloor x \right \rfloor=\begin{cases}0 & \text{ if } 0\leq x<1 \\ 1 & \text{ if } 1\leq x<2 \\ 2& \text{ if } 2\leq x<3 \\ ... & \text{ if } ...\\ n & \text{ if } n\leq x<n+1 \end{cases}`

In the given function is we have (x-1) instead of x, so we will get,

`\left \lfloor x-1 \right \rfloor=\begin{cases}0 & \text{ if } 1\leq x<2 \\ 1 & \text{ if } 2\leq x<3 \\ 2& \text{ if } 3\leq x<4 \\ ... & \text{ if } ...\\ n & \text{ if } n+1\leq x<n+2 \end{cases}`

It means the value of function is the integer n for all the number lies from (n+1) to (n+2), Where (n+2) is not included.

Similarly the value of function from 1 to 2, where 2 is excluded. Therefore the graph 4 represents the graph of the given function.