Which Of These Statements Is True For $f(x)=\left(\frac{9}{10}\right)^x$?A. The Domain Of $f(x$\] Is $x\ \textgreater \ 0$. B. The $y$-intercept Is $(0,1$\]. C. It Is Always Increasing. D. The Range Of

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Introduction

When dealing with exponential functions, it's essential to understand their properties, including domain, range, and behavior. In this article, we will analyze the function $f(x)=\left(\frac{9}{10}\right)^x$ and determine which of the given statements is true.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. For the function $f(x)=\left(\frac{9}{10}\right)^x$, we need to consider the restrictions on the input value $x$. Since the function involves a power with a base of $\frac{9}{10}$, which is a positive number, the function is defined for all real numbers. Therefore, the domain of the function is $x \in \mathbb{R}$.

Analyzing Statement A

Statement A claims that the domain of $f(x)$ is $x > 0$. However, as we have established earlier, the domain of the function is actually $x \in \mathbb{R}$, which includes all real numbers, both positive and negative. Therefore, statement A is false.

Finding the $y$-Intercept

The $y$-intercept of a function is the point where the graph of the function intersects the $y$-axis. To find the $y$-intercept of the function $f(x)=\left(\frac{9}{10}\right)^x$, we need to evaluate the function at $x=0$. Substituting $x=0$ into the function, we get:

$$f(0)=\left(\frac{9}{10}\right)^0=1$$

Therefore, the $y$-intercept of the function is $(0,1)$.

Analyzing Statement B

Statement B claims that the $y$-intercept is $(0,1)$. As we have just established, this is indeed the case. Therefore, statement B is true.

Behavior of the Function

To determine whether the function is always increasing, we need to examine its derivative. The derivative of the function $f(x)=\left(\frac{9}{10}\right)^x$ is given by:

$$f'(x)=\left(\frac{9}{10}\right)^x \ln\left(\frac{9}{10}\right)$$

Since the derivative is always positive, the function is always increasing.

Analyzing Statement C

Statement C claims that the function is always increasing. As we have established earlier, this is indeed the case. Therefore, statement C is true.

Range of the Function

The range of a function is the set of all possible output values. To determine the range of the function $f(x)=\left(\frac{9}{10}\right)^x$, we need to consider the behavior of the function as $x$ approaches positive and negative infinity. As $x$ approaches positive infinity, the function approaches 0, and as $x$ approaches negative infinity, the function approaches infinity. Therefore, the range of the function is $(0, \infty)$.

Analyzing Statement D

Statement D claims that the range of the function is $(0, \infty)$. As we have established earlier, this is indeed the case. Therefore, statement D is true.

Conclusion

In conclusion, we have analyzed the properties of the function $f(x)=\left(\frac{9}{10}\right)^x$ and determined that the following statements are true:

• The $y$-intercept is $(0,1)$.
• The function is always increasing.
• The range of the function is $(0, \infty)$.

Therefore, the correct answers are B, C, and D.

Final Thoughts

Understanding the properties of exponential functions is crucial in mathematics and other fields. By analyzing the domain, range, and behavior of the function $f(x)=\left(\frac{9}{10}\right)^x$, we have gained valuable insights into the properties of this function. This knowledge can be applied to a wide range of problems and is essential for success in mathematics and other fields.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Domain and Range of Exponential Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/funs.htm

Additional Resources

• Khan Academy: Exponential Functions
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Exponential Functions
  

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Introduction

In our previous article, we analyzed the properties of the function $f(x)=\left(\frac{9}{10}\right)^x$ and determined that the following statements are true:

• The $y$-intercept is $(0,1)$.
• The function is always increasing.
• The range of the function is $(0, \infty)$.

In this article, we will answer some frequently asked questions about the function $f(x)=\left(\frac{9}{10}\right)^x$.

Q: What is the domain of the function $f(x)=\left(\frac{9}{10}\right)^x$?

A: The domain of the function $f(x)=\left(\frac{9}{10}\right)^x$ is $x \in \mathbb{R}$, which includes all real numbers.

Q: What is the $y$-intercept of the function $f(x)=\left(\frac{9}{10}\right)^x$?

A: The $y$-intercept of the function $f(x)=\left(\frac{9}{10}\right)^x$ is $(0,1)$.

Q: Is the function $f(x)=\left(\frac{9}{10}\right)^x$ always increasing?

A: Yes, the function $f(x)=\left(\frac{9}{10}\right)^x$ is always increasing.

Q: What is the range of the function $f(x)=\left(\frac{9}{10}\right)^x$?

A: The range of the function $f(x)=\left(\frac{9}{10}\right)^x$ is $(0, \infty)$.

Q: How do I graph the function $f(x)=\left(\frac{9}{10}\right)^x$?

A: To graph the function $f(x)=\left(\frac{9}{10}\right)^x$, you can use a graphing calculator or software. The graph will be a curve that approaches the $y$-axis at $(0,1)$ and increases as $x$ increases.

Q: Can I use the function $f(x)=\left(\frac{9}{10}\right)^x$ to model real-world phenomena?

A: Yes, the function $f(x)=\left(\frac{9}{10}\right)^x$ 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 $f(x)=\left(\frac{9}{10}\right)^x$?

A: To find the derivative of the function $f(x)=\left(\frac{9}{10}\right)^x$, you can use the formula:

$$f'(x)=\left(\frac{9}{10}\right)^x \ln\left(\frac{9}{10}\right)$$

Q: Can I use the function $f(x)=\left(\frac{9}{10}\right)^x$ to solve optimization problems?

A: Yes, the function $f(x)=\left(\frac{9}{10}\right)^x$ can be used to solve optimization problems such as maximizing or minimizing a function.

Conclusion

In this article, we have answered some frequently asked questions about the function $f(x)=\left(\frac{9}{10}\right)^x$. We hope that this information has been helpful in understanding the properties and behavior of this function.

Final Thoughts

The function $f(x)=\left(\frac{9}{10}\right)^x$ is a simple yet powerful function that can be used to model a wide range of real-world phenomena. By understanding its properties and behavior, we can use it to solve optimization problems, model population growth, and analyze chemical reactions.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Domain and Range of Exponential Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/funs.htm

Additional Resources

• Khan Academy: Exponential Functions
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Exponential Functions