Graph The Function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$.Identify:1. Domain/Range2. Intercepts3. Asymptotes4. End Behavior5. Intervals Of Increasing/Decreasing

Introduction

In this article, we will explore the graph of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$. This function is an example of an exponential function, which is a type of function that has a constant base and a variable exponent. We will identify the domain and range of the function, find its intercepts, asymptotes, and end behavior, and determine the intervals where the function is increasing or decreasing.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$, the domain is all real numbers, since the function is defined for any value of $x$. The range of a function is the set of all possible output values. To find the range, we need to determine the possible values of $f(x)$.

Since the function is an exponential function with a base of $\frac{1}{2}$, which is less than 1, the function will approach 0 as $x$ approaches infinity. This means that the range of the function is all non-negative real numbers.

Intercepts

The intercepts of a function are the points where the function intersects the x-axis and the y-axis. To find the x-intercepts, we need to set $f(x) = 0$ and solve for $x$. Since the function is an exponential function, it will never intersect the x-axis, unless the base is 1, which is not the case here.

To find the y-intercept, we need to evaluate the function at $x = 0$. Plugging in $x = 0$ into the function, we get:

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

So, the y-intercept is $(0, 3)$.

Asymptotes

An asymptote is a line that the graph of a function approaches as the input values approach a certain value. In the case of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$, there is a horizontal asymptote at $y = 0$. This means that as $x$ approaches infinity, the function approaches 0.

End Behavior

The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity. In the case of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$, the end behavior is that the function approaches 0 as $x$ approaches infinity, and the function approaches positive infinity as $x$ approaches negative infinity.

Intervals of Increasing/Decreasing

To determine the intervals where the function is increasing or decreasing, we need to find the critical points of the function. The critical points are the points where the function changes from increasing to decreasing or from decreasing to increasing.

To find the critical points, we need to take the derivative of the function and set it equal to 0. The derivative of the function is:

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

Setting the derivative equal to 0, we get:

$3 \cdot \left(\frac{1}{2}\right)^x \cdot \ln\left(\frac{1}{2}\right) = 0$

Since the base is not 1, the derivative is never equal to 0. This means that the function is always increasing or decreasing.

To determine the intervals where the function is increasing or decreasing, we need to examine the sign of the derivative. Since the derivative is always positive, the function is always increasing.

Conclusion

In this article, we have explored the graph of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$. We have identified the domain and range of the function, found its intercepts, asymptotes, and end behavior, and determined the intervals where the function is increasing or decreasing. The function has a domain of all real numbers, a range of all non-negative real numbers, a y-intercept of $(0, 3)$, a horizontal asymptote at $y = 0$, and end behavior of approaching 0 as $x$ approaches infinity and approaching positive infinity as $x$ approaches negative infinity. The function is always increasing.

Graph

Here is a graph of the function:

# Graph of f(x) = 3 * (1/2)^x

## Domain and Range

* Domain: all real numbers
* Range: all non-negative real numbers

## Intercepts

* Y-intercept: (0, 3)

## Asymptotes

* Horizontal asymptote: y = 0

## End Behavior

* As x approaches infinity, f(x) approaches 0
* As x approaches negative infinity, f(x) approaches positive infinity

## Intervals of Increasing/Decreasing

* The function is always increasing

Frequently Asked Questions

Q: What is the domain of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The domain of the function is all real numbers.

Q: What is the range of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The range of the function is all non-negative real numbers.

Q: What is the y-intercept of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The y-intercept of the function is $(0, 3)$.

Q: What is the horizontal asymptote of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The horizontal asymptote of the function is $y = 0$.

Q: What is the end behavior of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: As $x$ approaches infinity, $f(x)$ approaches 0. As $x$ approaches negative infinity, $f(x)$ approaches positive infinity.

Q: Is the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$ always increasing or decreasing? A: The function is always increasing.

Q: How do I graph the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: To graph the function, you can use a graphing calculator or software. You can also use the information provided in this article to sketch the graph.

Q: What is the significance of the base $\frac{1}{2}$ in the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The base $\frac{1}{2}$ determines the rate at which the function approaches 0 as $x$ approaches infinity. A base less than 1 will cause the function to approach 0, while a base greater than 1 will cause the function to approach positive or negative infinity.

Q: Can I use the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$ to model real-world phenomena? A: Yes, the function can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.

Q: How do I find the derivative of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: To find the derivative, you can use the power rule and the chain rule. The derivative of the function is $f'(x) = 3 \cdot \left(\frac{1}{2}\right)^x \cdot \ln\left(\frac{1}{2}\right)$.

Q: What is the significance of the natural logarithm $\ln\left(\frac{1}{2}\right)$ in the derivative of the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$? A: The natural logarithm $\ln\left(\frac{1}{2}\right)$ determines the rate at which the function approaches 0 as $x$ approaches infinity. A natural logarithm less than 0 will cause the function to approach 0, while a natural logarithm greater than 0 will cause the function to approach positive or negative infinity.

Conclusion

In this article, we have provided answers to frequently asked questions about the function $f(x) = 3 \cdot \left(\frac{1}{2}\right)^x$. We have discussed the domain and range of the function, its intercepts, asymptotes, and end behavior, and its derivative. We have also provided information on how to graph the function and how to use it to model real-world phenomena.