Select The Correct Answer.Consider Function F F F : F ( X ) = { 2 X , X \textless 0 − X 2 − 4 X + 1 , 0 \textless X \textless 2 1 2 X + 3 , X \textgreater 2 F(x)=\left\{\begin{array}{ll} 2^x, & X\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ X\ \textless \ 2 \\ \frac{1}{2}x+3, & X\ \textgreater \ 2 \end{array}\right. F ( X ) = ⎩ ⎨ ⎧ ​ 2 X , − X 2 − 4 X + 1 , 2 1 ​ X + 3 , ​ X \textless 0 0 \textless X \textless 2 X \textgreater 2 ​ Which Statement Is

Introduction

In this article, we will delve into the world of mathematical functions and explore the given function $f$. The function $f$ is defined piecewise, meaning it has different definitions for different intervals of $x$. Our goal is to analyze the function and determine which statement is correct.

The Function $f$

The function $f$ is defined as:

$$f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$$

This function has three different definitions, each corresponding to a different interval of $x$. We will analyze each definition separately.

Definition 1: $2^x$ for $x < 0$

For $x < 0$, the function $f$ is defined as $2^x$. This is an exponential function, where the base is 2 and the exponent is $x$. As $x$ approaches negative infinity, $2^x$ approaches 0. As $x$ approaches 0 from the left, $2^x$ approaches 1.

Definition 2: $-x^2-4x+1$ for $0 < x < 2$

For $0 < x < 2$, the function $f$ is defined as $-x^2-4x+1$. This is a quadratic function, which is a polynomial of degree 2. The graph of this function is a parabola that opens downward. The vertex of the parabola is at $x = -2$, which is outside the interval $0 < x < 2$. Therefore, the function is decreasing over this interval.

Definition 3: $\frac{1}{2}x+3$ for $x > 2$

For $x > 2$, the function $f$ is defined as $\frac{1}{2}x+3$. This is a linear function, which is a polynomial of degree 1. The graph of this function is a straight line with a slope of $\frac{1}{2}$ and a y-intercept of 3.

Analyzing the Function

Now that we have analyzed each definition of the function $f$, we can examine the behavior of the function as a whole. We can see that the function has three different intervals, each with its own unique behavior.

• For $x < 0$, the function is an exponential function that approaches 0 as $x$ approaches negative infinity.
• For $0 < x < 2$, the function is a quadratic function that decreases over this interval.
• For $x > 2$, the function is a linear function that increases over this interval.

Conclusion

In conclusion, the function $f$ is a piecewise function that has three different definitions, each corresponding to a different interval of $x$. We have analyzed each definition separately and examined the behavior of the function as a whole. The function has an exponential definition for $x < 0$, a quadratic definition for $0 < x < 2$, and a linear definition for $x > 2$.

Which Statement is Correct?

Now that we have analyzed the function $f$, we can determine which statement is correct. The correct statement is:

• The function $f$ is a piecewise function that has three different definitions, each corresponding to a different interval of $x$.
• The function $f$ has an exponential definition for $x < 0$, a quadratic definition for $0 < x < 2$, and a linear definition for $x > 2$.
• The function $f$ approaches 0 as $x$ approaches negative infinity.
• The function $f$ decreases over the interval $0 < x < 2$.
• The function $f$ increases over the interval $x > 2$.

Final Answer

Introduction

In our previous article, we analyzed the given function $f$ and determined which statement is correct. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information about the function $f$.

Q: What is the domain of the function $f$?

A: The domain of the function $f$ is the set of all possible input values for $x$. Based on the given definitions, the domain of the function $f$ is $x < 0$, $0 < x < 2$, and $x > 2$.

Q: What is the range of the function $f$?

A: The range of the function $f$ is the set of all possible output values for $f(x)$. Based on the given definitions, the range of the function $f$ is $0 < f(x) < 1$ for $x < 0$, $-4 < f(x) < 1$ for $0 < x < 2$, and $f(x) > 3$ for $x > 2$.

Q: Is the function $f$ continuous?

A: The function $f$ is not continuous at $x = 0$ and $x = 2$. At $x = 0$, the function $f$ has a discontinuity because the left-hand limit and right-hand limit do not exist. At $x = 2$, the function $f$ has a discontinuity because the left-hand limit and right-hand limit do not exist.

Q: Is the function $f$ differentiable?

A: The function $f$ is not differentiable at $x = 0$ and $x = 2$. At $x = 0$, the function $f$ has a cusp because the left-hand derivative and right-hand derivative do not exist. At $x = 2$, the function $f$ has a cusp because the left-hand derivative and right-hand derivative do not exist.

Q: What is the value of the function $f$ at $x = -1$?

A: To find the value of the function $f$ at $x = -1$, we need to use the definition of the function $f$ for $x < 0$. The definition of the function $f$ for $x < 0$ is $f(x) = 2^x$. Therefore, the value of the function $f$ at $x = -1$ is $f(-1) = 2^{-1} = \frac{1}{2}$.

Q: What is the value of the function $f$ at $x = 1$?

A: To find the value of the function $f$ at $x = 1$, we need to use the definition of the function $f$ for $0 < x < 2$. The definition of the function $f$ for $0 < x < 2$ is $f(x) = -x^2 - 4x + 1$. Therefore, the value of the function $f$ at $x = 1$ is $f(1) = -(1)^2 - 4(1) + 1 = -4$.

Q: What is the value of the function $f$ at $x = 3$?

A: To find the value of the function $f$ at $x = 3$, we need to use the definition of the function $f$ for $x > 2$. The definition of the function $f$ for $x > 2$ is $f(x) = \frac{1}{2}x + 3$. Therefore, the value of the function $f$ at $x = 3$ is $f(3) = \frac{1}{2}(3) + 3 = 4.5$.

Conclusion

In this Q&A article, we have provided additional information about the function $f$ and clarified any doubts. We have discussed the domain and range of the function $f$, its continuity and differentiability, and the value of the function $f$ at specific points. We hope that this article has been helpful in understanding the function $f$.

Final Answer

The final answer is: The function $f$ is a piecewise function that has three different definitions, each corresponding to a different interval of $x$.