Given The Function F ( X ) = 3 ( 1 2 ) X F(x) = 3\left(\frac{1}{2}\right)^x F ( X ) = 3 ( 2 1 ​ ) X , Complete The Table And Determine The Characteristics Of The Function.$[ \begin{array}{|c|c|} \hline x & Y \ \hline -2 & 3 \times \left(\frac{1}{2}\right)^{-2} \ \hline -1 & 3

The given function is $f(x) = 3\left(\frac$1}{2}\right)^x3(21​)x. This is an exponential function with a base of $\frac{1{2}$ and a coefficient of $3$. The function is defined for all real values of $x$.

Completing the Table

To complete the table, we need to calculate the values of $y$ for different values of $x$. We will use the given function to calculate the values of $y$.

x	y
-2	3 × (1/2)^(-2)
-1	3
0	3 × (1/2)^0
1	3 × (1/2)^1
2	3 × (1/2)^2

Let's calculate the values of $y$ for each value of $x$.

Calculating y for x = -2

To calculate the value of $y$ for $x = -2$, we substitute $x = -2$ into the function:

$f(-2) = 3\left(\frac{1}{2}\right)^{-2}$

Using the property of exponents that $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as:

$f(-2) = 3\left(\frac{2}{1}\right)^2$

Simplifying the expression, we get:

$f(-2) = 3 \times 4 = 12$

So, the value of $y$ for $x = -2$ is $12$.

Calculating y for x = -1

To calculate the value of $y$ for $x = -1$, we substitute $x = -1$ into the function:

$f(-1) = 3\left(\frac{1}{2}\right)^{-1}$

Using the property of exponents that $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as:

$f(-1) = 3\left(\frac{2}{1}\right)$

Simplifying the expression, we get:

$f(-1) = 3 \times 2 = 6$

So, the value of $y$ for $x = -1$ is $6$.

Calculating y for x = 0

To calculate the value of $y$ for $x = 0$, we substitute $x = 0$ into the function:

$f(0) = 3\left(\frac{1}{2}\right)^0$

Using the property of exponents that $a^0 = 1$, we can rewrite the expression as:

$f(0) = 3 \times 1 = 3$

So, the value of $y$ for $x = 0$ is $3$.

Calculating y for x = 1

To calculate the value of $y$ for $x = 1$, we substitute $x = 1$ into the function:

$f(1) = 3\left(\frac{1}{2}\right)^1$

Simplifying the expression, we get:

$f(1) = 3 \times \frac{1}{2} = \frac{3}{2}$

So, the value of $y$ for $x = 1$ is $\frac{3}{2}$.

Calculating y for x = 2

To calculate the value of $y$ for $x = 2$, we substitute $x = 2$ into the function:

$f(2) = 3\left(\frac{1}{2}\right)^2$

Simplifying the expression, we get:

$f(2) = 3 \times \frac{1}{4} = \frac{3}{4}$

So, the value of $y$ for $x = 2$ is $\frac{3}{4}$.

Updated Table

x	y
-2	12
-1	6
0	3
1	1.5
2	0.75

Characteristics of the Function

The function $f(x) = 3\left(\frac{1}{2}\right)^x$ is an exponential function with a base of $\frac{1}{2}$ and a coefficient of $3$. The function is defined for all real values of $x$.

Domain of the Function

The domain of the function is all real numbers, i.e., $x \in (-\infty, \infty)$.

Range of the Function

The range of the function is all positive real numbers, i.e., $y \in (0, \infty)$.

End Behavior of the Function

As $x$ approaches negative infinity, $y$ approaches infinity.

As $x$ approaches positive infinity, $y$ approaches $0$.

Intercepts of the Function

The function has a $y$-intercept at $(0, 3)$.

The function has no $x$-intercepts.

Asymptotes of the Function

The function has a horizontal asymptote at $y = 0$.

The function has no vertical asymptotes.

Increasing and Decreasing Intervals

The function is decreasing on the interval $(-\infty, \infty)$.

Local Maxima and Minima

The function has no local maxima or minima.

Concavity of the Function

The function is concave down on the interval $(-\infty, \infty)$.

Inflection Points

The function has no inflection points.

Second Derivative

The second derivative of the function is:

$f''(x) = 3\left(\frac{1}{2}\right)^x \ln\left(\frac{1}{2}\right)^2$

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In the previous article, we explored the function $f(x) = 3\left(\frac{1}{2}\right)^x$ and completed a table to determine its characteristics. In this article, we will answer some frequently asked questions about the function and its characteristics.

Q: What is the domain of the function?

A: The domain of the function is all real numbers, i.e., $x \in (-\infty, \infty)$.

Q: What is the range of the function?

A: The range of the function is all positive real numbers, i.e., $y \in (0, \infty)$.

Q: How does the function behave as x approaches negative infinity?

A: As $x$ approaches negative infinity, $y$ approaches infinity.

Q: How does the function behave as x approaches positive infinity?

A: As $x$ approaches positive infinity, $y$ approaches $0$.

Q: What is the y-intercept of the function?

A: The function has a $y$-intercept at $(0, 3)$.

Q: Does the function have any x-intercepts?

A: No, the function has no $x$-intercepts.

Q: Does the function have any vertical asymptotes?

A: No, the function has no vertical asymptotes.

Q: Is the function increasing or decreasing?

A: The function is decreasing on the interval $(-\infty, \infty)$.

Q: Does the function have any local maxima or minima?

A: No, the function has no local maxima or minima.

Q: Is the function concave up or concave down?

A: The function is concave down on the interval $(-\infty, \infty)$.

Q: Does the function have any inflection points?

A: No, the function has no inflection points.

Q: How do you calculate the second derivative of the function?

A: The second derivative of the function is:

$f''(x) = 3\left(\frac{1}{2}\right)^x \ln\left(\frac{1}{2}\right)^2$

Q: What does the second derivative tell us about the function?

A: The second derivative is negative for all values of $x$, indicating that the function is concave down on the interval $(-\infty, \infty)$.

Q: How can you use the function to model real-world situations?

A: The function can be used to model situations where a quantity decreases exponentially over time. For example, the function can be used to model the decay of a radioactive substance or the decrease in population of a species over time.

Q: How can you graph the function?

A: The function can be graphed using a graphing calculator or a computer algebra system. The graph of the function will be a decreasing exponential curve that approaches the x-axis as x approaches positive infinity.

Q: What are some common applications of the function?

A: The function has many applications in fields such as physics, engineering, and economics. Some common applications of the function include:

• Modeling the decay of radioactive substances
• Modeling the decrease in population of a species over time
• Modeling the growth of a population over time
• Modeling the decay of a chemical reaction
• Modeling the growth of a financial investment over time

We hope this Q&A article has helped you understand the function and its characteristics. If you have any further questions, please don't hesitate to ask.