Which Points Lie On The Graph Of The Function $f(x) = [x] + 2$? Check All That Apply.- $(-5.5, -4$\]- $(-3.8, -2$\]- $(-1.1, 1$\]- $(-0.9, 2$\]- $(2.2, 5$\]- $(4.7, 6$\]

Understanding the Graph of the Function $f(x) = [x] + 2$

The graph of a function is a visual representation of the relationship between the input values (x) and the output values (y). In this case, we are given the function $f(x) = [x] + 2$, where $[x]$ represents the greatest integer less than or equal to $x$. To determine which points lie on the graph of this function, we need to evaluate the function at each given point and check if the resulting value matches the y-coordinate of the point.

Evaluating the Function at Each Given Point

Let's start by evaluating the function at each given point:

$(-5.5, -4)$

To evaluate the function at $x = -5.5$, we need to find the greatest integer less than or equal to $-5.5$, which is $-6$. Then, we add $2$ to get:

$f(-5.5) = [-5.5] + 2 = -6 + 2 = -4$

Since the resulting value matches the y-coordinate of the point, this point lies on the graph of the function.

$(-3.8, -2)$

To evaluate the function at $x = -3.8$, we need to find the greatest integer less than or equal to $-3.8$, which is $-4$. Then, we add $2$ to get:

$f(-3.8) = [-3.8] + 2 = -4 + 2 = -2$

Since the resulting value matches the y-coordinate of the point, this point lies on the graph of the function.

$(-1.1, 1)$

To evaluate the function at $x = -1.1$, we need to find the greatest integer less than or equal to $-1.1$, which is $-2$. Then, we add $2$ to get:

$f(-1.1) = [-1.1] + 2 = -2 + 2 = 0$

Since the resulting value does not match the y-coordinate of the point, this point does not lie on the graph of the function.

$(-0.9, 2)$

To evaluate the function at $x = -0.9$, we need to find the greatest integer less than or equal to $-0.9$, which is $-1$. Then, we add $2$ to get:

$f(-0.9) = [-0.9] + 2 = -1 + 2 = 1$

Since the resulting value does not match the y-coordinate of the point, this point does not lie on the graph of the function.

$(2.2, 5)$

To evaluate the function at $x = 2.2$, we need to find the greatest integer less than or equal to $2.2$, which is $2$. Then, we add $2$ to get:

$f(2.2) = [2.2] + 2 = 2 + 2 = 4$

Since the resulting value does not match the y-coordinate of the point, this point does not lie on the graph of the function.

$(4.7, 6)$

To evaluate the function at $x = 4.7$, we need to find the greatest integer less than or equal to $4.7$, which is $4$. Then, we add $2$ to get:

$f(4.7) = [4.7] + 2 = 4 + 2 = 6$

Since the resulting value matches the y-coordinate of the point, this point lies on the graph of the function.

Conclusion

In conclusion, the points that lie on the graph of the function $f(x) = [x] + 2$ are:

• $(-5.5, -4)$
• $(-3.8, -2)$
• $(4.7, 6)$

These points satisfy the equation of the function, and their y-coordinates match the resulting values of the function evaluated at the corresponding x-coordinates.
Q&A: Understanding the Graph of the Function $f(x) = [x] + 2$

In our previous article, we explored the graph of the function $f(x) = [x] + 2$ and determined which points lie on the graph. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What is the greatest integer function $[x]$?

A: The greatest integer function $[x]$ is a function that returns the greatest integer less than or equal to $x$. For example, $[3.7] = 3$, $[-2.3] = -3$, and $[0] = 0$.

Q: How does the function $f(x) = [x] + 2$ work?

A: The function $f(x) = [x] + 2$ works by first finding the greatest integer less than or equal to $x$, and then adding $2$ to the result. For example, if $x = 3.7$, then $[x] = 3$, and $f(x) = 3 + 2 = 5$.

Q: What is the domain of the function $f(x) = [x] + 2$?

A: The domain of the function $f(x) = [x] + 2$ is all real numbers, since the greatest integer function $[x]$ is defined for all real numbers.

Q: What is the range of the function $f(x) = [x] + 2$?

A: The range of the function $f(x) = [x] + 2$ is all integers, since the greatest integer function $[x]$ returns an integer, and adding $2$ to an integer results in another integer.

Q: How does the graph of the function $f(x) = [x] + 2$ behave for negative values of $x$?

A: For negative values of $x$, the graph of the function $f(x) = [x] + 2$ behaves in a step-like manner, with each step occurring at an integer value of $x$. For example, if $x = -3.7$, then $[x] = -4$, and $f(x) = -4 + 2 = -2$.

Q: How does the graph of the function $f(x) = [x] + 2$ behave for positive values of $x$?

A: For positive values of $x$, the graph of the function $f(x) = [x] + 2$ behaves in a step-like manner, with each step occurring at an integer value of $x$. For example, if $x = 3.7$, then $[x] = 3$, and $f(x) = 3 + 2 = 5$.

Q: Can the graph of the function $f(x) = [x] + 2$ be represented as a continuous function?

A: No, the graph of the function $f(x) = [x] + 2$ cannot be represented as a continuous function, since the greatest integer function $[x]$ is a step-like function that has discontinuities at integer values of $x$.

Q: Can the graph of the function $f(x) = [x] + 2$ be represented as a piecewise function?

A: Yes, the graph of the function $f(x) = [x] + 2$ can be represented as a piecewise function, with each piece corresponding to a different interval of $x$.

Conclusion

In conclusion, the graph of the function $f(x) = [x] + 2$ is a step-like function that behaves in a predictable manner for both positive and negative values of $x$. The function has a domain of all real numbers and a range of all integers. While the graph cannot be represented as a continuous function, it can be represented as a piecewise function.