Consider The Graph Of The Function $f(x)=\left(\frac{1}{4}\right)^x$.Which Statements Describe Key Features Of Function $f$?- Domain: $\{x \mid X \in \mathbb{R}\}$- Horizontal Asymptote: $y=0$- $y$-intercept

Exploring Key Features of the Function $f(x)=\left(\frac{1}{4}\right)^x$

The given function $f(x)=\left(\frac{1}{4}\right)^x$ is an exponential function with a base of $\frac{1}{4}$. In this article, we will delve into the key features of this function, including its domain, horizontal asymptote, and $y$-intercept.

Domain of the Function

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 $f(x)=\left(\frac{1}{4}\right)^x$, the base is a positive real number, and the exponent is any real number. This means that the function is defined for all real numbers, and the domain is given by:

Domain: $\{x \mid x \in \mathbb{R}\}$

This indicates that the function is defined for all real numbers, including positive and negative numbers, as well as zero.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input values become very large, either positively or negatively. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$, we can analyze the behavior of the function as $x$ becomes very large.

As $x$ becomes very large, the value of $\left(\frac{1}{4}\right)^x$ approaches zero. This is because the base $\frac{1}{4}$ is less than 1, and raising a number less than 1 to a large power results in a value that approaches zero.

Horizontal Asymptote: $y=0$

This indicates that the graph of the function approaches the horizontal line $y=0$ as $x$ becomes very large.

$y$-Intercept

The $y$-intercept of a function is the point where the graph of the function intersects the $y$-axis. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$, we can find the $y$-intercept by evaluating the function at $x=0$.

$y$-Intercept: $f(0)=\left(\frac{1}{4}\right)^0=1$

This indicates that the graph of the function intersects the $y$-axis at the point $(0,1)$.

Graph of the Function

The graph of the function $f(x)=\left(\frac{1}{4}\right)^x$ is a decreasing exponential curve that approaches the horizontal line $y=0$ as $x$ becomes very large. The graph passes through the point $(0,1)$, which is the $y$-intercept.

Graph of the Function:

The graph of the function $f(x)=\left(\frac{1}{4}\right)^x$ is a decreasing exponential curve that approaches the horizontal line $y=0$ as $x$ becomes very large.

Conclusion

In conclusion, the function $f(x)=\left(\frac{1}{4}\right)^x$ has a domain of all real numbers, a horizontal asymptote of $y=0$, and a $y$-intercept of $(0,1)$. The graph of the function is a decreasing exponential curve that approaches the horizontal line $y=0$ as $x$ becomes very large.

Key Takeaways:

• The domain of the function $f(x)=\left(\frac{1}{4}\right)^x$ is all real numbers.
• The horizontal asymptote of the function is $y=0$.
• The $y$-intercept of the function is $(0,1)$.
• The graph of the function is a decreasing exponential curve that approaches the horizontal line $y=0$ as $x$ becomes very large.

Final Thoughts:

The function $f(x)=\left(\frac{1}{4}\right)^x$ is a simple yet powerful example of an exponential function. By analyzing its key features, we can gain a deeper understanding of the behavior of exponential functions and their applications in various fields.
Frequently Asked Questions about the Function $f(x)=\left(\frac{1}{4}\right)^x$

In the previous article, we explored the key features of the function $f(x)=\left(\frac{1}{4}\right)^x$, including its domain, horizontal asymptote, and $y$-intercept. In this article, we will answer some frequently asked questions about this function.

Q: What is the domain of the function $f(x)=\left(\frac{1}{4}\right)^x$?

A: The domain of the function $f(x)=\left(\frac{1}{4}\right)^x$ is all real numbers, denoted by $\{x \mid x \in \mathbb{R}\}$. This means that the function is defined for all real numbers, including positive and negative numbers, as well as zero.

Q: What is the horizontal asymptote of the function $f(x)=\left(\frac{1}{4}\right)^x$?

A: The horizontal asymptote of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $y=0$. This means that as $x$ becomes very large, the value of $\left(\frac{1}{4}\right)^x$ approaches zero.

Q: What is the $y$-intercept of the function $f(x)=\left(\frac{1}{4}\right)^x$?

A: The $y$-intercept of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $(0,1)$. This means that the graph of the function intersects the $y$-axis at the point $(0,1)$.

Q: How does the function $f(x)=\left(\frac{1}{4}\right)^x$ behave as $x$ becomes very large?

A: As $x$ becomes very large, the value of $\left(\frac{1}{4}\right)^x$ approaches zero. This is because the base $\frac{1}{4}$ is less than 1, and raising a number less than 1 to a large power results in a value that approaches zero.

Q: Can the function $f(x)=\left(\frac{1}{4}\right)^x$ be used to model real-world phenomena?

A: Yes, the function $f(x)=\left(\frac{1}{4}\right)^x$ can be used to model real-world phenomena, such as population growth or decay, chemical reactions, and electrical circuits. The function's ability to approach zero as $x$ becomes very large makes it a useful tool for modeling situations where a quantity approaches zero over time.

Q: How can the function $f(x)=\left(\frac{1}{4}\right)^x$ be used in finance?

A: The function $f(x)=\left(\frac{1}{4}\right)^x$ can be used in finance to model the growth or decay of an investment over time. For example, if an investment grows at a rate of 25% per year, the function $f(x)=\left(\frac{1}{4}\right)^x$ can be used to model the investment's growth over time.

Q: Can the function $f(x)=\left(\frac{1}{4}\right)^x$ be used to model the spread of diseases?

A: Yes, the function $f(x)=\left(\frac{1}{4}\right)^x$ can be used to model the spread of diseases. The function's ability to approach zero as $x$ becomes very large makes it a useful tool for modeling situations where a disease spreads rapidly at first, but then slows down over time.

Q: How can the function $f(x)=\left(\frac{1}{4}\right)^x$ be used in science?

A: The function $f(x)=\left(\frac{1}{4}\right)^x$ can be used in science to model the decay of radioactive materials, the growth of populations, and the spread of diseases. The function's ability to approach zero as $x$ becomes very large makes it a useful tool for modeling situations where a quantity approaches zero over time.

Q: Can the function $f(x)=\left(\frac{1}{4}\right)^x$ be used to model the behavior of electrical circuits?

A: Yes, the function $f(x)=\left(\frac{1}{4}\right)^x$ can be used to model the behavior of electrical circuits. The function's ability to approach zero as $x$ becomes very large makes it a useful tool for modeling situations where a quantity approaches zero over time.

Conclusion

In conclusion, the function $f(x)=\left(\frac{1}{4}\right)^x$ is a simple yet powerful example of an exponential function. By answering some frequently asked questions about this function, we can gain a deeper understanding of its behavior and its applications in various fields.

Key Takeaways:

• The domain of the function $f(x)=\left(\frac{1}{4}\right)^x$ is all real numbers.
• The horizontal asymptote of the function is $y=0$.
• The $y$-intercept of the function is $(0,1)$.
• The function approaches zero as $x$ becomes very large.
• The function can be used to model real-world phenomena, such as population growth or decay, chemical reactions, and electrical circuits.

Final Thoughts:

The function $f(x)=\left(\frac{1}{4}\right)^x$ is a useful tool for modeling situations where a quantity approaches zero over time. By understanding its behavior and its applications, we can gain a deeper understanding of the world around us.