Mathematics, 02.12.2019 16:31, EllaLovesAnime

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

f(x)= [x]-3

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

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answered: marrizza7

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 .

If is an integer than [x] is also the same integer and if 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 in the interval [-3,-2) the value of the function F(x)=[x]-3 is .

Substitute x=-2 in equation (1).

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

Substitute x=-1.5 in equation .

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

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

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 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 .

The value of the function for different value of 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 is which is incorrect because as per the table 1 (attached in the end) the value of the function in the interval is .

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 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 .

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 in the interval $-3\leq x$ is , in the interval $-2\leq x$ is , in the interval $-1\leq x$ is 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 .

But from the graph in option D it is observed that for all value of in the interval $-3\leq x\leq -2$ the value of the function is 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

2.A problem to find radius and center of circle

3.A problem to determine intercepts of a line

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.