Select All The Correct Answers.Consider The Graph Of The Function F ( X ) = ( 1 4 ) X F(x)=\left(\frac{1}{4}\right)^x F ( X ) = ( 4 1 ) X .Which Statements Describe Key Features Of The Function?- Domain Of { X ∣ − 1 \textless X \textless ∞ } \{x \mid -1 \ \textless \ X \ \textless \ \infty\} { X ∣ − 1 \textless X \textless ∞ } -

Mar 12, 2025 by ADMIN 337 views

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

The function $f(x)=\left(\frac{1}{4}\right)^x$ is an exponential function with a base of $\frac{1}{4}$ . This function has several key features that are essential to understand in order to analyze and graph the function. In this article, we will discuss the domain, range, and other key features of the function $f(x)=\left(\frac{1}{4}\right)^x$ .

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 domain is given as $\{x \mid -1 \ \textless \ x \ \textless \ \infty\}$ . This means that the function is defined for all real numbers greater than -1.

The range of a function is the set of all possible output values for which the function is defined. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$ , the range is given as $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$ . This means that the function is defined for all real numbers greater than 0.

An asymptote is a line that the graph of a function approaches as the input values get arbitrarily large. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$ , there is a horizontal asymptote at $y=0$ . This means that as the input values get arbitrarily large, the output values approach 0.

An intercept is a point where the graph of a function intersects with the x-axis or the y-axis. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$ , there is a y-intercept at $(0,1)$ . This means that the graph of the function intersects with the y-axis at the point (0,1).

An increasing interval is an interval where the function is increasing, and a decreasing interval is an interval where the function is decreasing. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$ , the function is increasing on the interval $(-1,\infty)$ and decreasing on the interval $(-\infty,-1)$ .

Based on the above discussion, the key features of the function $f(x)=\left(\frac{1}{4}\right)^x$ are:

Domain: $\{x \mid -1 \ \textless \ x \ \textless \ \infty\}$
Range: $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$
Horizontal Asymptote: $y=0$
Y-Intercept: $(0,1)$
Increasing Interval: $(-1,\infty)$
Decreasing Interval: $(-\infty,-1)$

In conclusion, the function $f(x)=\left(\frac{1}{4}\right)^x$ has several key features that are essential to understand in order to analyze and graph the function. The domain, range, asymptotes, intercepts, and increasing and decreasing intervals are all important features of the function. By understanding these key features, we can better analyze and graph the function.

The final answer is:

Domain: $\{x \mid -1 \ \textless \ x \ \textless \ \infty\}$
Range: $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$
Horizontal Asymptote: $y=0$
Y-Intercept: $(0,1)$
Increasing Interval: $(-1,\infty)$
Decreasing Interval: $(-\infty,-1)$
Q&A: Understanding Key Features of the Function $f(x)=\left(\frac{1}{4}\right)^x$

In our previous article, we discussed the key features of the function $f(x)=\left(\frac{1}{4}\right)^x$ . In this article, we will answer some frequently asked questions about the function and its key features.

A: The domain of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $\{x \mid -1 \ \textless \ x \ \textless \ \infty\}$ . This means that the function is defined for all real numbers greater than -1.

A: The range of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$ . This means that the function is defined for all real numbers greater than 0.

A: The horizontal asymptote of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $y=0$ . This means that as the input values get arbitrarily large, the output values approach 0.

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 with the y-axis at the point (0,1).

A: The increasing interval of the function $f(x)=\left(\frac{1}{4}\right)^x$ is $(-1,\infty)$ , and the decreasing interval is $(-\infty,-1)$ .

A: To graph the function $f(x)=\left(\frac{1}{4}\right)^x$ , you can start by plotting the y-intercept at (0,1). Then, use the increasing and decreasing intervals to determine the direction of the graph. Finally, use the horizontal asymptote to determine the behavior of the graph as the input values get arbitrarily large.

A: The function $f(x)=\left(\frac{1}{4}\right)^x$ has several real-world applications, including:

Modeling population growth and decline
Modeling the spread of diseases
Modeling the behavior of electrical circuits
Modeling the behavior of financial systems

In conclusion, the function $f(x)=\left(\frac{1}{4}\right)^x$ has several key features that are essential to understand in order to analyze and graph the function. By understanding the domain, range, asymptotes, intercepts, and increasing and decreasing intervals, we can better analyze and graph the function. Additionally, the function has several real-world applications, including modeling population growth and decline, modeling the spread of diseases, modeling the behavior of electrical circuits, and modeling the behavior of financial systems.

The final answer is:

Domain: $\{x \mid -1 \ \textless \ x \ \textless \ \infty\}$
Range: $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$
Horizontal Asymptote: $y=0$
Y-Intercept: $(0,1)$
Increasing Interval: $(-1,\infty)$
Decreasing Interval: $(-\infty,-1)$