Which Points Lie On The Graph Of The Function F ( X ) = ⌈ X ⌉ + 2 F(x) = \lceil X \rceil + 2 F ( X ) = ⌈ X ⌉ + 2 ? Check All That Apply.A. ( − 5.5 , − 4 (-5.5, -4 ( − 5.5 , − 4 ]B. ( − 3.8 , − 2 (-3.8, -2 ( − 3.8 , − 2 ]C. ( − 1.1 , 1 (-1.1, 1 ( − 1.1 , 1 ]D. ( − 0.9 , 2 (-0.9, 2 ( − 0.9 , 2 ]E. ( 2.2 , 5 (2.2, 5 ( 2.2 , 5 ]F. ( 4.7 , 6 (4.7, 6 ( 4.7 , 6 ]

The function $f(x) = \lceil x \rceil + 2$ involves the ceiling function, which rounds a number up to the nearest integer. This function is often denoted by $\lceil x \rceil$. To understand the graph of this function, we need to consider how the ceiling function affects the input values of $x$.

The Ceiling Function and Its Effect on $x$

The ceiling function $\lceil x \rceil$ takes any real number $x$ and rounds it up to the nearest integer. For example, $\lceil 3.7 \rceil = 4$, $\lceil -2.3 \rceil = -2$, and $\lceil 0 \rceil = 0$. This means that the function $f(x) = \lceil x \rceil + 2$ will also be affected by the ceiling function.

Evaluating the Function $f(x) = \lceil x \rceil + 2$

To evaluate the function $f(x) = \lceil x \rceil + 2$, we need to consider the possible values of $x$ and how the ceiling function affects them. Let's consider some examples:

• For $x = -3.8$, we have $\lceil -3.8 \rceil = -3$, so $f(-3.8) = -3 + 2 = -1$.
• For $x = -1.1$, we have $\lceil -1.1 \rceil = -1$, so $f(-1.1) = -1 + 2 = 1$.
• For $x = 2.2$, we have $\lceil 2.2 \rceil = 3$, so $f(2.2) = 3 + 2 = 5$.
• For $x = 4.7$, we have $\lceil 4.7 \rceil = 5$, so $f(4.7) = 5 + 2 = 7$.

Checking the Answer Choices

Now that we have a better understanding of the function $f(x) = \lceil x \rceil + 2$, let's check the answer choices:

A. $(-5.5, -4)$: Since $\lceil -5.5 \rceil = -5$, we have $f(-5.5) = -5 + 2 = -3$. This point does not lie on the graph of the function.

B. $(-3.8, -2)$: Since $\lceil -3.8 \rceil = -3$, we have $f(-3.8) = -3 + 2 = -1$. This point does not lie on the graph of the function.

C. $(-1.1, 1)$: Since $\lceil -1.1 \rceil = -1$, we have $f(-1.1) = -1 + 2 = 1$. This point lies on the graph of the function.

D. $(-0.9, 2)$: Since $\lceil -0.9 \rceil = -0$, we have $f(-0.9) = -0 + 2 = 2$. This point does not lie on the graph of the function.

E. $(2.2, 5)$: Since $\lceil 2.2 \rceil = 3$, we have $f(2.2) = 3 + 2 = 5$. This point lies on the graph of the function.

F. $(4.7, 6)$: Since $\lceil 4.7 \rceil = 5$, we have $f(4.7) = 5 + 2 = 7$. This point does not lie on the graph of the function.

Conclusion

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

• $(-1.1, 1)$
• $(2.2, 5)$

Q: What is the ceiling function, and how does it affect the input values of $x$?

A: The ceiling function $\lceil x \rceil$ takes any real number $x$ and rounds it up to the nearest integer. For example, $\lceil 3.7 \rceil = 4$, $\lceil -2.3 \rceil = -2$, and $\lceil 0 \rceil = 0$. This means that the function $f(x) = \lceil x \rceil + 2$ will also be affected by the ceiling function.

Q: How do you evaluate the function $f(x) = \lceil x \rceil + 2$?

A: To evaluate the function $f(x) = \lceil x \rceil + 2$, you need to consider the possible values of $x$ and how the ceiling function affects them. Let's consider some examples:

• For $x = -3.8$, we have $\lceil -3.8 \rceil = -3$, so $f(-3.8) = -3 + 2 = -1$.
• For $x = -1.1$, we have $\lceil -1.1 \rceil = -1$, so $f(-1.1) = -1 + 2 = 1$.
• For $x = 2.2$, we have $\lceil 2.2 \rceil = 3$, so $f(2.2) = 3 + 2 = 5$.
• For $x = 4.7$, we have $\lceil 4.7 \rceil = 5$, so $f(4.7) = 5 + 2 = 7$.

Q: How do you check if a point lies on the graph of the function $f(x) = \lceil x \rceil + 2$?

A: To check if a point lies on the graph of the function $f(x) = \lceil x \rceil + 2$, you need to evaluate the function at the given point and see if it satisfies the equation. Let's consider some examples:

• For the point $(-1.1, 1)$, we have $\lceil -1.1 \rceil = -1$, so $f(-1.1) = -1 + 2 = 1$. This point lies on the graph of the function.
• For the point $(2.2, 5)$, we have $\lceil 2.2 \rceil = 3$, so $f(2.2) = 3 + 2 = 5$. This point lies on the graph of the function.

Q: What are the points that lie on the graph of the function $f(x) = \lceil x \rceil + 2$?

A: The points that lie on the graph of the function $f(x) = \lceil x \rceil + 2$ are:

• $(-1.1, 1)$
• $(2.2, 5)$

These points satisfy the equation $f(x) = \lceil x \rceil + 2$, and they lie on the graph of the function.

Q: How does the ceiling function affect the graph of the function $f(x) = \lceil x \rceil + 2$?

A: The ceiling function affects the graph of the function $f(x) = \lceil x \rceil + 2$ by rounding the input values of $x$ up to the nearest integer. This means that the graph of the function will have a series of horizontal line segments, each corresponding to a different integer value of $x$.

Q: What is the significance of the points that lie on the graph of the function $f(x) = \lceil x \rceil + 2$?

A: The points that lie on the graph of the function $f(x) = \lceil x \rceil + 2$ are significant because they satisfy the equation $f(x) = \lceil x \rceil + 2$. This means that these points are the only points that lie on the graph of the function, and they are the only points that satisfy the equation.

Q: How can you use the graph of the function $f(x) = \lceil x \rceil + 2$ to solve problems?

A: You can use the graph of the function $f(x) = \lceil x \rceil + 2$ to solve problems by evaluating the function at different points and seeing if they lie on the graph. For example, you can use the graph to find the value of the function at a given point, or to determine if a point lies on the graph of the function.