Question

Which graph represents the function on the interval [-3 , 3] ? f(x)= [x]-3

Answer 1

Answer:

the graph that best corresponds to the function in the given interval is the  down left graph.

Step-by-step explanation:

Given : f(x)= [x]-3 and graph .

To find  : Which graph represents the function on the interval [-3 , 3] ?

Solution :  Let's evaluate the extremes of the interval in given function f(x).

X= -3,             f( -3 )= [ -3 ] - 3 .

                      f( -3 ) = -6 .

X= 3,          f( 3 )= [ 3 ] - 3 .

                      f( 3 ) = 0.

Therefore, the graph that best corresponds to the function in the given interval is the  down left graph.

Answer 2

The graph in the $\fbox{\begin\\\ \bf option C\\\end{minisapce}}$ represents the function $F(x)=[x]-3$

Further explanation:

The function is given as follows:

$F(x)=[x]-3$

Label the above equation as follows:

$F(x)=[x]-3$                                          (1)

In the above equation $[x]$ represents a greatest integer function.

Greatest integer function $[x]$ is a function which is an integral part of any real number $x$.

If $x$ is an integer than $[x]$ is also the same integer and if $x$ is in decimals than the value of $[x]$ is the next smallest integer.

For example: If $x=-2.5$ than $[x]=-3$ and if $x=-3$ than $[x]=-3$

Figure 1 (attached in the end) indicates the graph of the function $y=[x]$.

The function is $F(x)=[x]-3$ and the interval is given as $[-3,3]$.

Substitute $x=-3$ in equation (1).

$\begin{aligned}F(-3)&=[-3]-3\\&=-3-3\\&=-6\end{aligned}$

Substitute $x=-2.5$ in equation (1).

$\begin{aligned}F(-2.5)&=[-2.5]-3\\&=-3-3\\&=-6\end{aligned}$

From the above calculation it is concluded that for each value of $x$ in the interval $[-3,-2)$ the value of the function $F(x)=[x]-3$ is $-6$.

Substitute $x=-2$ in equation (1).

$\begin{aligned}F(-2)&=[-2]-3\\&=-2-3\\&=-5\end{aligned}$

Substitute $x=-1.5$ in equation $1$.

$\begin{aligned}F(-1.5)&=[-1.5]-3\\&=-2-3\\&=-5\end{aligned}$

From the above calculation it is concluded that for each value of $x$ in the interval $[-2,-1)$ the value of the function $F(x)=[x]-3$ is $-5$.

Similarly, the value of the function $F(x)=[x]-3$ in the interval $[-1,0), [0,1), [1,2)$ and $[2,3)$ are $-4, -3, -2$ and $-1$ respectively.

Substitute $x=3$ in equation (1).

$\begin{aligned}F(3)&=[3]-3\\&=3-3\\&=0\end{aligned}$

Only for $x=3$ the value of the function $F(x)=[x]-3$ is $0$.

The value of the function for different value of $x$ lying in different intervals is given in the table attached in the end of the solution.

Option A:

From the graph given in the option A it is observed that the value of the function $F(x)=[x]-3$ in the interval $-3$ is $-5$which is incorrect because as per the table 1 (attached in the end) the value of the function in the interval $-3$ is $-6$.

This implies that the option A is incorrect.

Option B:

From the graph given in the option B it is observed that the value of the function $F(x)=[x]-3$ in the interval $-3\leq x$ is $-5$ which is incorrect because as per the table 1 (attached in the end) the value of the function in the interval $-3\leq x$ is $-6$.

This implies that the option B is incorrect.

Option C:

From the graph given in the option C it is observed that the value of the function $F(x)=[x]-3$ for all the values of $x$ in the interval $-3\leq x$ is $-6$, in the interval $-2\leq x$ is $-5$, in the interval $-1\leq x$ is $-4$ and so on.

The value of the function as observed from the graph in option C for different intervals is exactly same as the calculation made above.

This implies that option C is correct.

Option D:

As per the our calculation the value of the function $F(x)=[x]-3$ in the interval $-3\leq x$ is $-6$  .

But from the graph in option D it is observed that for all value of $x$ in the interval $-3\leq x\leq -2$ the value of the function is $-5$ which is not correct.

This implies that option D is incorrect.

Therefore, the graph in the $\fbox{\begin\\\ \bf option C\\\end{minisapce}}$ represents the function $F(x)=[x]-3$.

Learn more:

1. A problem on composite function brainly.com/question/2723982  

2. A problem to find radius and center of circle brainly.com/question/9510228  

3. A problem to determine intercepts of a line brainly.com/question/1332667  

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Functions

Keywords: Functions, greatest integer function, floor function, domain , range, intervals, equation, graph, curve, relation, F(x)=[x]-3, [x], smallest integer function, ceiling function.