Identify The Key Features Of The Graph Of The Function $f(x) = \log_2(x-1) + 2$.- Vertical Asymptote At $x = 1$- Horizontal Asymptote At $y = 1$- No $x$-intercept- No $y$-intercept- Domain Of $(1,

Understanding the Graph of the Function $f(x) = \log_2(x-1) + 2$

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this article, we will focus on identifying the key features of the graph of the function $f(x) = \log_2(x-1) + 2$. This function involves a logarithmic function with base 2, and we will explore its properties to determine the characteristics of its graph.

Key Features of the Graph

Vertical Asymptote at $x = 1$

The function $f(x) = \log_2(x-1) + 2$ has a vertical asymptote at $x = 1$. This means that as $x$ approaches 1 from the right, the value of $f(x)$ increases without bound. In other words, the function approaches positive infinity as $x$ approaches 1 from the right. This is because the logarithmic function is undefined when its argument is less than or equal to 0, and in this case, $x-1$ is less than 0 when $x$ is less than 1.

To understand why this is the case, let's consider the definition of a logarithmic function. The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to produce $x$. In this case, we have $\log_2(x-1) = y$, which means that $2^y = x-1$. Since $x-1$ is less than 0 when $x$ is less than 1, we have $2^y < 0$, which is not possible since $2^y$ is always positive. Therefore, the function is undefined when $x$ is less than 1, and we have a vertical asymptote at $x = 1$.

Horizontal Asymptote at $y = 1$

The function $f(x) = \log_2(x-1) + 2$ has a horizontal asymptote at $y = 1$. This means that as $x$ approaches infinity, the value of $f(x)$ approaches 1. In other words, the function approaches 1 as $x$ increases without bound.

To understand why this is the case, let's consider the behavior of the logarithmic function as its argument increases without bound. As $x$ approaches infinity, $x-1$ also approaches infinity, and therefore $\log_2(x-1)$ approaches infinity. However, since we are adding 2 to the logarithmic function, the value of $f(x)$ approaches 1 as $x$ approaches infinity.

No $x$-intercept

The function $f(x) = \log_2(x-1) + 2$ does not have an $x$-intercept. This means that there is no value of $x$ for which $f(x) = 0$. In other words, the function does not cross the $x$-axis.

To understand why this is the case, let's consider the definition of an $x$-intercept. An $x$-intercept is a point on the graph of a function where the $y$-coordinate is 0. In this case, we have $f(x) = \log_2(x-1) + 2 = 0$, which implies that $\log_2(x-1) = -2$. However, this is not possible since the logarithmic function is always positive, and therefore we do not have an $x$-intercept.

No $y$-intercept

The function $f(x) = \log_2(x-1) + 2$ does not have a $y$-intercept. This means that there is no value of $y$ for which $f(x) = y$ when $x = 0$. In other words, the function does not cross the $y$-axis.

To understand why this is the case, let's consider the definition of a $y$-intercept. A $y$-intercept is a point on the graph of a function where the $x$-coordinate is 0. In this case, we have $f(0) = \log_2(0-1) + 2 = \log_2(-1) + 2$. However, this is not possible since the logarithmic function is undefined when its argument is negative, and therefore we do not have a $y$-intercept.

Domain of $(1, \infty)$

The domain of the function $f(x) = \log_2(x-1) + 2$ is $(1, \infty)$. This means that the function is defined for all values of $x$ greater than 1.

To understand why this is the case, let's consider the definition of the domain of a function. The domain of a function is the set of all possible input values for which the function is defined. In this case, we have $x-1 > 0$ when $x > 1$, and therefore the function is defined for all values of $x$ greater than 1.

In conclusion, the graph of the function $f(x) = \log_2(x-1) + 2$ has a vertical asymptote at $x = 1$, a horizontal asymptote at $y = 1$, no $x$-intercept, no $y$-intercept, and a domain of $(1, \infty)$. These characteristics are a result of the properties of the logarithmic function and the specific form of the function.
Q&A: Understanding the Graph of the Function $f(x) = \log_2(x-1) + 2$

In our previous article, we explored the key features of the graph of the function $f(x) = \log_2(x-1) + 2$. We discussed the vertical asymptote at $x = 1$, the horizontal asymptote at $y = 1$, the lack of $x$-intercept and $y$-intercept, and the domain of $(1, \infty)$. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What is the significance of the vertical asymptote at $x = 1$?

A: The vertical asymptote at $x = 1$ indicates that the function approaches positive infinity as $x$ approaches 1 from the right. This means that the function is undefined at $x = 1$, and the graph of the function will have a vertical line at $x = 1$.

Q: Why does the function have a horizontal asymptote at $y = 1$?

A: The horizontal asymptote at $y = 1$ indicates that the function approaches 1 as $x$ approaches infinity. This is because the logarithmic function grows slowly as its argument increases, and the addition of 2 to the logarithmic function shifts the graph up by 2 units.

Q: Why does the function not have an $x$-intercept?

A: The function does not have an $x$-intercept because the logarithmic function is always positive, and therefore the function $f(x) = \log_2(x-1) + 2$ is always greater than 0. This means that the graph of the function will never cross the $x$-axis.

Q: Why does the function not have a $y$-intercept?

A: The function does not have a $y$-intercept because the logarithmic function is undefined when its argument is negative, and therefore the function $f(x) = \log_2(x-1) + 2$ is undefined at $x = 0$. This means that the graph of the function will never cross the $y$-axis.

Q: What is the domain of the function?

A: The domain of the function is $(1, \infty)$. This means that the function is defined for all values of $x$ greater than 1.

Q: How can I graph the function?

A: To graph the function, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the function. To create a table of values, you can choose a few values of $x$ greater than 1 and calculate the corresponding values of $f(x)$.

Q: What are some real-world applications of this function?

A: This function has several real-world applications, including:

• Modeling population growth: The function can be used to model the growth of a population over time, where the population grows at a rate proportional to the current population.
• Modeling chemical reactions: The function can be used to model the rate of a chemical reaction, where the rate of reaction is proportional to the concentration of the reactants.
• Modeling electrical circuits: The function can be used to model the behavior of an electrical circuit, where the voltage across a resistor is proportional to the current flowing through it.

Q: Can I use this function to model other phenomena?

A: Yes, you can use this function to model other phenomena, such as:

• Modeling the growth of a tumor: The function can be used to model the growth of a tumor over time, where the tumor grows at a rate proportional to the current size of the tumor.
• Modeling the behavior of a pendulum: The function can be used to model the behavior of a pendulum, where the angle of the pendulum is proportional to the time elapsed.
• Modeling the behavior of a spring: The function can be used to model the behavior of a spring, where the displacement of the spring is proportional to the force applied to it.

In conclusion, the graph of the function $f(x) = \log_2(x-1) + 2$ has several key features, including a vertical asymptote at $x = 1$, a horizontal asymptote at $y = 1$, no $x$-intercept, no $y$-intercept, and a domain of $(1, \infty)$. These characteristics make the function useful for modeling a wide range of phenomena, including population growth, chemical reactions, and electrical circuits.