Select The Correct Texts In The Table.Consider Function F F F .$ f(x)=\left{ \begin{array}{ll} -\frac{1}{4} X^2+6 X+36, & X\ \textless \ -2 \ 4 X-15, & -2 \leq X\ \textless \ 4 \ 3^{x-4}, & X\ \textgreater \

Selecting the Correct Texts in the Table: A Mathematical Analysis

In this article, we will delve into the world of mathematics and explore a function $f$ defined by a piecewise function. The function $f$ is defined as:

$$f(x)=\left\{ \begin{array}{ll} -\frac{1}{4} x^2+6 x+36, & x\ \textless \ -2 \\ 4 x-15, & -2 \leq x\ \textless \ 4 \\ 3^{x-4}, & x\ \textgreater \ \end{array} \right.$$

Our goal is to select the correct texts in the table that correspond to the function $f$ for different values of $x$. We will analyze each part of the function and determine which text is correct for each value of $x$.

For $x < -2$, the function $f$ is defined as:

$$f(x) = -\frac{1}{4} x^2 + 6x + 36$$

This is a quadratic function, and we can analyze its behavior by looking at its graph. The graph of this function is a parabola that opens downward, with a vertex at $x = -6$. Since $x < -2$, the function is always decreasing for $x < -2$.

For $-2 \leq x < 4$, the function $f$ is defined as:

$$f(x) = 4x - 15$$

This is a linear function, and we can analyze its behavior by looking at its graph. The graph of this function is a straight line with a slope of 4 and a y-intercept of -15. Since $-2 \leq x < 4$, the function is always increasing for $-2 \leq x < 4$.

For $x > 4$, the function $f$ is defined as:

$$f(x) = 3^{x-4}$$

This is an exponential function, and we can analyze its behavior by looking at its graph. The graph of this function is an exponential curve that increases rapidly as $x$ increases. Since $x > 4$, the function is always increasing for $x > 4$.

In conclusion, we have analyzed the function $f$ and determined which text is correct for each value of $x$. For $x < -2$, the correct text is $-\frac{1}{4} x^2 + 6x + 36$. For $-2 \leq x < 4$, the correct text is $4x - 15$. For $x > 4$, the correct text is $3^{x-4}$.

The function $f$ is a piecewise function that is defined differently for different values of $x$. We have analyzed each part of the function and determined which text is correct for each value of $x$. This type of function is commonly used in mathematics to model real-world phenomena that have different behaviors in different regions.

1. Find the value of $f(-3)$.
2. Find the value of $f(2)$.
3. Find the value of $f(5)$.

1. For $x = -3$, we have $x < -2$, so the correct text is $-\frac{1}{4} x^2 + 6x + 36$. Plugging in $x = -3$, we get:

$$f(-3) = -\frac{1}{4} (-3)^2 + 6(-3) + 36 = -\frac{1}{4} (9) - 18 + 36 = -\frac{9}{4} - 18 + 36 = -\frac{9}{4} + 18 = -\frac{9}{4} + \frac{72}{4} = \frac{63}{4}$$

2. For $x = 2$, we have $-2 \leq x < 4$, so the correct text is $4x - 15$. Plugging in $x = 2$, we get:

$$f(2) = 4(2) - 15 = 8 - 15 = -7$$

3. For $x = 5$, we have $x > 4$, so the correct text is $3^{x-4}$. Plugging in $x = 5$, we get:

$$f(5) = 3^{5-4} = 3^1 = 3$$

In conclusion, we have analyzed the function $f$ and determined which text is correct for each value of $x$. We have also solved example problems to demonstrate how to use the function. This type of function is commonly used in mathematics to model real-world phenomena that have different behaviors in different regions.
Q&A: Selecting the Correct Texts in the Table

In our previous article, we analyzed the function $f$ and determined which text is correct for each value of $x$. We also solved example problems to demonstrate how to use the function. In this article, we will answer some frequently asked questions about the function $f$ and provide additional examples to help you understand the concept better.

A: The domain of the function $f$ is the set of all possible values of $x$ for which the function is defined. In this case, the domain of the function $f$ is $x < -2$, $-2 \leq x < 4$, and $x > 4$.

A: To determine which text is correct for a given value of $x$, you need to check which part of the domain the value of $x$ falls into. If $x < -2$, the correct text is $-\frac{1}{4} x^2 + 6x + 36$. If $-2 \leq x < 4$, the correct text is $4x - 15$. If $x > 4$, the correct text is $3^{x-4}$.

A: Yes, the function $f$ can be used to model real-world phenomena that have different behaviors in different regions. For example, the function $f$ can be used to model the growth of a population that has different growth rates in different regions.

A: To graph the function $f$, you need to graph each part of the function separately. For $x < -2$, the graph of the function is a parabola that opens downward. For $-2 \leq x < 4$, the graph of the function is a straight line with a slope of 4. For $x > 4$, the graph of the function is an exponential curve that increases rapidly.

A: Yes, the function $f$ can be used to solve equations. For example, you can use the function $f$ to solve the equation $f(x) = 10$.

A: To solve the equation $f(x) = 10$, you need to substitute the value of $f(x)$ into the equation and solve for $x$. For $x < -2$, the equation becomes $-\frac{1}{4} x^2 + 6x + 36 = 10$. For $-2 \leq x < 4$, the equation becomes $4x - 15 = 10$. For $x > 4$, the equation becomes $3^{x-4} = 10$.

A: Yes, the function $f$ can be used to model systems of equations. For example, you can use the function $f$ to model a system of two equations with two variables.

A: To model a system of two equations with two variables using the function $f$, you need to define two functions $f_1(x)$ and $f_2(x)$ that satisfy the two equations. Then, you can use the function $f$ to model the system of equations.

In conclusion, we have answered some frequently asked questions about the function $f$ and provided additional examples to help you understand the concept better. We have also shown how to use the function $f$ to model real-world phenomena, graph the function, solve equations, and model systems of equations.