Consider The Graph Of The Function F ( X ) = 10 X F(x)=10^x F ( X ) = 1 0 X .What Is The Range Of The Function G ( X ) = − 2 F ( X ) + 1 G(x)=-2f(x)+1 G ( X ) = − 2 F ( X ) + 1 ?

Introduction

When dealing with functions, it's essential to understand how transformations affect their behavior and characteristics. In this article, we will explore the transformation of the function $f(x)=10^x$ to find the range of the function $g(x)=-2f(x)+1$. We will delve into the properties of exponential functions, transformations, and how they impact the range of a function.

The Original Function $f(x)=10^x$

The function $f(x)=10^x$ is an exponential function with base 10. This function has a few key properties that are essential to understand:

• Domain: The domain of $f(x)$ is all real numbers, denoted as $(-\infty, \infty)$.
• Range: The range of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$.
• Asymptotes: The function has a horizontal asymptote at $y=0$ and a vertical asymptote at $x=-\infty$.

The Transformation of $f(x)$ to $g(x)$

The function $g(x)$ is a transformation of $f(x)$, where $g(x)=-2f(x)+1$. This transformation involves three steps:

1. Vertical Stretch: The function $f(x)$ is stretched vertically by a factor of 2, resulting in $-2f(x)$.
2. Reflection: The function $-2f(x)$ is reflected across the x-axis, resulting in $-2f(x)$.
3. Vertical Shift: The function $-2f(x)$ is shifted vertically by 1 unit, resulting in $-2f(x)+1$.

The Properties of $g(x)$

After applying the transformation, the function $g(x)$ has the following properties:

• Domain: The domain of $g(x)$ is the same as the domain of $f(x)$, which is all real numbers, denoted as $(-\infty, \infty)$.
• Range: The range of $g(x)$ is not immediately apparent, as it depends on the transformation applied to $f(x)$.
• Asymptotes: The function $g(x)$ has a horizontal asymptote at $y=1$ and a vertical asymptote at $x=-\infty$.

Finding the Range of $g(x)$

To find the range of $g(x)$, we need to consider the transformation applied to $f(x)$. The function $g(x)$ is a vertical stretch and reflection of $f(x)$, followed by a vertical shift. This means that the range of $g(x)$ will be a transformation of the range of $f(x)$.

The range of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$. When we apply a vertical stretch and reflection to $f(x)$, the range becomes all negative real numbers, denoted as $(-\infty, 0)$. Finally, when we apply a vertical shift of 1 unit, the range becomes all real numbers less than or equal to 1, denoted as $(-\infty, 1]$.

Conclusion

In conclusion, the range of the function $g(x)=-2f(x)+1$ is all real numbers less than or equal to 1, denoted as $(-\infty, 1]$. This is a result of the transformation applied to the original function $f(x)=10^x$, which involved a vertical stretch, reflection, and vertical shift.

Final Thoughts

Understanding the transformation of functions is crucial in mathematics, as it allows us to analyze and manipulate functions in various ways. By applying transformations to functions, we can create new functions with different properties and characteristics. In this article, we explored the transformation of the function $f(x)=10^x$ to find the range of the function $g(x)=-2f(x)+1$. We hope that this article has provided valuable insights into the properties of exponential functions and transformations.

Introduction

In our previous article, we explored the transformation of the function $f(x)=10^x$ to find the range of the function $g(x)=-2f(x)+1$. We discussed the properties of exponential functions, transformations, and how they impact the range of a function. In this article, we will answer some frequently asked questions (FAQs) about the transformation of the function $f(x)=10^x$.

Q: What is the domain of the function $f(x)=10^x$?

A: The domain of the function $f(x)=10^x$ is all real numbers, denoted as $(-\infty, \infty)$.

Q: What is the range of the function $f(x)=10^x$?

A: The range of the function $f(x)=10^x$ is all positive real numbers, denoted as $(0, \infty)$.

Q: What is the horizontal asymptote of the function $f(x)=10^x$?

A: The horizontal asymptote of the function $f(x)=10^x$ is $y=0$.

Q: What is the vertical asymptote of the function $f(x)=10^x$?

A: The vertical asymptote of the function $f(x)=10^x$ is $x=-\infty$.

Q: How does the transformation $g(x)=-2f(x)+1$ affect the domain of the function $f(x)=10^x$?

A: The transformation $g(x)=-2f(x)+1$ does not affect the domain of the function $f(x)=10^x$, which remains all real numbers, denoted as $(-\infty, \infty)$.

Q: How does the transformation $g(x)=-2f(x)+1$ affect the range of the function $f(x)=10^x$?

A: The transformation $g(x)=-2f(x)+1$ affects the range of the function $f(x)=10^x$, which becomes all real numbers less than or equal to 1, denoted as $(-\infty, 1]$.

Q: What is the horizontal asymptote of the function $g(x)=-2f(x)+1$?

A: The horizontal asymptote of the function $g(x)=-2f(x)+1$ is $y=1$.

Q: What is the vertical asymptote of the function $g(x)=-2f(x)+1$?

A: The vertical asymptote of the function $g(x)=-2f(x)+1$ is $x=-\infty$.

Q: Can the transformation $g(x)=-2f(x)+1$ be applied to any function $f(x)$?

A: No, the transformation $g(x)=-2f(x)+1$ can only be applied to functions that have a range of all positive real numbers.

Q: What is the significance of the transformation $g(x)=-2f(x)+1$ in real-world applications?

A: The transformation $g(x)=-2f(x)+1$ has various applications in real-world scenarios, such as modeling population growth, chemical reactions, and financial transactions.

Conclusion

In conclusion, the transformation of the function $f(x)=10^x$ to find the range of the function $g(x)=-2f(x)+1$ is a crucial concept in mathematics. By understanding the properties of exponential functions and transformations, we can analyze and manipulate functions in various ways. We hope that this article has provided valuable insights into the FAQs about the transformation of the function $f(x)=10^x$.

Final Thoughts

Understanding the transformation of functions is essential in mathematics, as it allows us to create new functions with different properties and characteristics. By applying transformations to functions, we can model real-world scenarios and make predictions about the behavior of complex systems. In this article, we explored the transformation of the function $f(x)=10^x$ to find the range of the function $g(x)=-2f(x)+1$. We hope that this article has provided a comprehensive understanding of the transformation of the function $f(x)=10^x$.