The Function $f(x) = \log_2 X$ Is Replaced By $f\left(x - \frac{5}{6}\right$\]. How Does The Domain Change?A. The Domain Of $f(x$\] Is $x \ \textgreater \ 0$, While The Domain Of $f\left(x -

The Function Shift: Understanding the Impact on the Domain of $f(x) = \log_2 x$

When a function undergoes a transformation, its domain may also change. In this article, we will explore how the domain of the function $f(x) = \log_2 x$ changes when it is replaced by $f\left(x - \frac{5}{6}\right)$. We will delve into the concept of function shifts, the properties of logarithmic functions, and how these transformations affect the domain of the function.

The Original Function: $f(x) = \log_2 x$

The original function $f(x) = \log_2 x$ is a logarithmic function with base 2. The domain of this function is all positive real numbers, denoted as $x > 0$. This means that the function is defined for all values of $x$ greater than 0.

The Shifted Function: $f\left(x - \frac{5}{6}\right)$

The shifted function $f\left(x - \frac{5}{6}\right)$ is obtained by replacing $x$ with $x - \frac{5}{6}$ in the original function. This transformation shifts the graph of the function to the right by $\frac{5}{6}$ units.

Understanding the Shift

To understand the impact of the shift on the domain, let's consider the point $(x, f(x))$ on the graph of the original function. When we replace $x$ with $x - \frac{5}{6}$, the new point becomes $\left(x - \frac{5}{6}, f\left(x - \frac{5}{6}\right)\right)$.

The New Domain

Since the graph of the function is shifted to the right by $\frac{5}{6}$ units, the new domain of the function is all values of $x$ greater than $\frac{5}{6}$. In other words, the domain of the shifted function is $x > \frac{5}{6}$.

Comparison of the Domains

To summarize, the domain of the original function $f(x) = \log_2 x$ is $x > 0$, while the domain of the shifted function $f\left(x - \frac{5}{6}\right)$ is $x > \frac{5}{6}$. This means that the shifted function is defined for all values of $x$ greater than $\frac{5}{6}$, which is a subset of the domain of the original function.

In conclusion, when the function $f(x) = \log_2 x$ is replaced by $f\left(x - \frac{5}{6}\right)$, the domain changes from $x > 0$ to $x > \frac{5}{6}$. This transformation shifts the graph of the function to the right by $\frac{5}{6}$ units, resulting in a new domain that is a subset of the original domain. Understanding the impact of function shifts on the domain is essential in mathematics, and this article provides a clear explanation of this concept.

1. If the function $f(x) = \log_3 x$ is replaced by $f\left(x - \frac{2}{3}\right)$, what is the new domain of the function?
2. If the function $f(x) = \log_2 x$ is replaced by $f\left(x + \frac{1}{2}\right)$, what is the new domain of the function?
3. If the function $f(x) = \log_5 x$ is replaced by $f\left(x - \frac{3}{5}\right)$, what is the new domain of the function?

1. The new domain of the function is $x > \frac{2}{3}$.
2. The new domain of the function is $x > -\frac{1}{2}$.
3. The new domain of the function is $x > \frac{2}{5}$.

In this article, we explored how the domain of the function $f(x) = \log_2 x$ changes when it is replaced by $f\left(x - \frac{5}{6}\right)$. We discussed the concept of function shifts, the properties of logarithmic functions, and how these transformations affect the domain of the function. By understanding the impact of function shifts on the domain, we can better analyze and solve problems involving logarithmic functions.
Q&A: Understanding the Impact of Function Shifts on the Domain of Logarithmic Functions

In our previous article, we explored how the domain of the function $f(x) = \log_2 x$ changes when it is replaced by $f\left(x - \frac{5}{6}\right)$. We discussed the concept of function shifts, the properties of logarithmic functions, and how these transformations affect the domain of the function. In this article, we will answer some frequently asked questions about function shifts and their impact on the domain of logarithmic functions.

Q1: What is a function shift?

A function shift is a transformation that moves the graph of a function to the left or right by a certain number of units. In the case of the function $f(x) = \log_2 x$, a shift to the right by $\frac{5}{6}$ units results in the new function $f\left(x - \frac{5}{6}\right)$.

Q2: How does a function shift affect the domain of a logarithmic function?

A function shift affects the domain of a logarithmic function by changing the values of $x$ for which the function is defined. In the case of the function $f(x) = \log_2 x$, a shift to the right by $\frac{5}{6}$ units results in a new domain of $x > \frac{5}{6}$.

Q3: What is the relationship between the original domain and the new domain after a function shift?

The original domain and the new domain after a function shift are related in that the new domain is a subset of the original domain. In the case of the function $f(x) = \log_2 x$, the original domain is $x > 0$, while the new domain after a shift to the right by $\frac{5}{6}$ units is $x > \frac{5}{6}$.

Q4: How do I determine the new domain after a function shift?

To determine the new domain after a function shift, you need to consider the shift value and the original domain. In the case of the function $f(x) = \log_2 x$, a shift to the right by $\frac{5}{6}$ units results in a new domain of $x > \frac{5}{6}$.

Q5: Can a function shift result in a change in the range of a logarithmic function?

A function shift can result in a change in the range of a logarithmic function. However, the range of a logarithmic function is typically all real numbers, and a function shift does not change this property.

Q6: How do I apply function shifts to other types of functions?

Function shifts can be applied to other types of functions, such as polynomial and rational functions. However, the impact of a function shift on the domain of these functions may be different from the impact on the domain of a logarithmic function.

Q7: What are some common applications of function shifts in mathematics and other fields?

Function shifts have many applications in mathematics and other fields, such as physics and engineering. They are used to model real-world phenomena, such as population growth and chemical reactions.

Q8: How can I visualize a function shift?

A function shift can be visualized by graphing the original function and the shifted function on the same coordinate plane. This allows you to see the impact of the shift on the domain and range of the function.

In conclusion, function shifts have a significant impact on the domain of logarithmic functions. By understanding how function shifts affect the domain, you can better analyze and solve problems involving logarithmic functions. We hope that this Q&A article has provided you with a better understanding of function shifts and their applications in mathematics and other fields.

1. If the function $f(x) = \log_3 x$ is replaced by $f\left(x - \frac{2}{3}\right)$, what is the new domain of the function?
2. If the function $f(x) = \log_2 x$ is replaced by $f\left(x + \frac{1}{2}\right)$, what is the new domain of the function?
3. If the function $f(x) = \log_5 x$ is replaced by $f\left(x - \frac{3}{5}\right)$, what is the new domain of the function?

1. The new domain of the function is $x > \frac{2}{3}$.
2. The new domain of the function is $x > -\frac{1}{2}$.
3. The new domain of the function is $x > \frac{2}{5}$.

In this article, we answered some frequently asked questions about function shifts and their impact on the domain of logarithmic functions. We hope that this Q&A article has provided you with a better understanding of function shifts and their applications in mathematics and other fields.