Consider The Graph Of The Function $f(x)=\left(\frac{1}{4}\right)^x$.Which Statements Describe Key Features Of The Function $f$?A. $x$-intercept At (3,0) B. Domain Of $\{x \mid -1 \ \textless \ X \ \textless \

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

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this case, we are given the function $f(x)=\left(\frac{1}{4}\right)^x$. To understand the key features of this function, we need to analyze its behavior and identify its domain, range, and any notable points on the graph.

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 input value $x$ can take on any real number. However, since the function involves exponentiation, we need to consider the restrictions on the input value.

The base of the exponent, $\frac{1}{4}$, is a positive number, and the exponent, $x$, can take on any real number. Therefore, the domain of the function is all real numbers, or in interval notation, $(-\infty, \infty)$.

Range of the Function

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 output value is always positive, since the base of the exponent is positive and the exponent is a real number.

As $x$ increases, the output value of the function decreases, and as $x$ decreases, the output value of the function increases. Therefore, the range of the function is all positive real numbers, or in interval notation, $(0, \infty)$.

Asymptotes

An asymptote is a line that the graph of a function approaches as the input value gets arbitrarily close to a certain point. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$, there is a horizontal asymptote at $y=0$.

As $x$ approaches negative infinity, the output value of the function approaches 0, but never reaches it. Therefore, the horizontal asymptote at $y=0$ is a natural boundary for the function.

X-Intercept

The x-intercept of a function is the point on the graph where the output value is 0. In the case of the function $f(x)=\left(\frac{1}{4}\right)^x$, the x-intercept is not at the point (3,0), since the function is always positive and never reaches 0.

However, the x-intercept is at the point (0,1), since the output value of the function is 1 when the input value is 0.

Key Features of the Function

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

• Domain: all real numbers, or in interval notation, $(-\infty, \infty)$
• Range: all positive real numbers, or in interval notation, $(0, \infty)$
• Horizontal asymptote: $y=0$
• X-intercept: (0,1)

Therefore, the correct statements that describe key features of the function $f$ are:

• Domain of $\{x \mid -\infty \ \textless \ x \ \textless \ \infty\}$
• Range of $\{y \mid 0 \ \textless \ y \ \textless \ \infty\}$
• Horizontal asymptote at $y=0$
• X-intercept at (0,1)

The statement "x-intercept at (3,0)" is incorrect, since the x-intercept is at the point (0,1), not (3,0).
Q&A: Understanding the Graph of the Function $f(x)=\left(\frac{1}{4}\right)^x$

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

Q: What is the behavior of the function as x approaches negative infinity?

A: As x approaches negative infinity, the output value of the function approaches 0, but never reaches it. This is because the base of the exponent, $\frac{1}{4}$, is a positive number, and the exponent, $x$, is a negative number. As x gets more negative, the output value of the function gets closer and closer to 0, but it never actually reaches 0.

Q: What is the behavior of the function as x approaches positive infinity?

A: As x approaches positive infinity, the output value of the function approaches 0, but never reaches it. This is because the base of the exponent, $\frac{1}{4}$, is a positive number, and the exponent, $x$, is a positive number. As x gets larger and larger, the output value of the function gets smaller and smaller, approaching 0, but never actually reaching it.

Q: Is the function continuous?

A: Yes, the function $f(x)=\left(\frac{1}{4}\right)^x$ is continuous for all real numbers. This means that the function has no gaps or jumps in its graph, and it can be evaluated at any point in its domain.

Q: Is the function differentiable?

A: Yes, the function $f(x)=\left(\frac{1}{4}\right)^x$ is differentiable for all real numbers. This means that the function has a well-defined derivative at every point in its domain, and it can be used to find the slope of the tangent line to the graph at any point.

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

A: The x-intercept of the function $f(x)=\left(\frac{1}{4}\right)^x$ is the point (0,1). This is because the output value of the function is 1 when the input value is 0.

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

A: The y-intercept of the function $f(x)=\left(\frac{1}{4}\right)^x$ is the point (0,1). This is because the output value of the function is 1 when the input value is 0.

Q: Is the function increasing or decreasing?

A: The function $f(x)=\left(\frac{1}{4}\right)^x$ is decreasing for all real numbers. This means that as the input value x increases, the output value of the function decreases.

Q: Is the function concave up or concave down?

A: The function $f(x)=\left(\frac{1}{4}\right)^x$ is concave down for all real numbers. This means that the graph of the function is curved downward, and the slope of the tangent line to the graph decreases as the input value x increases.

Q: What is the equation of the horizontal asymptote?

A: The equation of the horizontal asymptote of the function $f(x)=\left(\frac{1}{4}\right)^x$ is y=0. This means that as the input value x approaches negative infinity or positive infinity, the output value of the function approaches 0.

Q: What is the equation of the vertical asymptote?

A: There is no vertical asymptote for the function $f(x)=\left(\frac{1}{4}\right)^x$. This means that the graph of the function does not have any vertical lines that the graph approaches as the input value x gets arbitrarily close to a certain point.

Q: Is the function periodic?

A: No, the function $f(x)=\left(\frac{1}{4}\right)^x$ is not periodic. This means that the graph of the function does not repeat itself at regular intervals, and it does not have any periodic behavior.

Q: Is the function symmetric?

A: No, the function $f(x)=\left(\frac{1}{4}\right)^x$ is not symmetric. This means that the graph of the function does not have any symmetry about the x-axis or the y-axis.

Q: What is the graph of the function like?

A: The graph of the function $f(x)=\left(\frac{1}{4}\right)^x$ is a curve that approaches the x-axis as the input value x approaches negative infinity or positive infinity. The graph is concave down and has a horizontal asymptote at y=0. The x-intercept is at the point (0,1), and the y-intercept is also at the point (0,1).