Select The Correct Answer.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 G G If G ( X ) = − F ( X ) − 5 G(x)=-f(x)-5 G ( X ) = − F ( X ) − 5 ?A. { Y ∣ − ∞ \textless Y \textless ∞ } \{y \mid -\infty \ \textless \ Y \ \textless \ \infty\} { Y ∣ − ∞ \textless Y \textless ∞ } B. ${y \mid -5 \

Introduction

When dealing with functions, it's essential to understand how transformations affect their graphs. In this article, we'll explore the concept of function transformation and how it applies to the given problem. We'll delve into the world of exponential functions, specifically the function $f(x) = 10^x$, and examine how the function $g(x) = -f(x) - 5$ is related to it.

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

The function $f(x) = 10^x$ is an exponential function with a base of 10. This function is known for its rapid growth as $x$ increases. To understand the behavior of this function, let's consider its graph. The graph of $f(x) = 10^x$ is a continuous, increasing curve that passes through the point (0, 1). As $x$ increases, the value of $f(x)$ also increases, and as $x$ decreases, the value of $f(x)$ decreases.

The Transformed Function: $g(x) = -f(x) - 5$

Now, let's consider the function $g(x) = -f(x) - 5$. This function is a transformation of the original function $f(x) = 10^x$. The negative sign in front of $f(x)$ indicates a reflection across the x-axis, while the constant term -5 indicates a vertical shift downward by 5 units.

Understanding the Reflection

When a function is reflected across the x-axis, its graph is flipped upside down. This means that if the original function $f(x)$ has a value of $y$ at a given point $x$, the reflected function $-f(x)$ will have a value of $-y$ at the same point $x$. In the case of the function $f(x) = 10^x$, the reflection across the x-axis will result in a graph that is a mirror image of the original graph.

Understanding the Vertical Shift

A vertical shift is a transformation that moves the graph of a function up or down by a certain number of units. In the case of the function $g(x) = -f(x) - 5$, the vertical shift is downward by 5 units. This means that if the original function $f(x)$ has a value of $y$ at a given point $x$, the transformed function $g(x)$ will have a value of $y - 5$ at the same point $x$.

The Range of the Function $g$

Now that we've understood the transformation of the function $f(x) = 10^x$ to $g(x) = -f(x) - 5$, let's determine the range of the function $g$. The range of a function is the set of all possible output values it can produce for the given input values.

Analyzing the Reflection

Since the function $g(x)$ is a reflection of the function $f(x)$ across the x-axis, the range of $g(x)$ will be the negative of the range of $f(x)$. The range of $f(x) = 10^x$ is all positive real numbers, i.e., $\{y \mid y > 0\}$. Therefore, the range of $g(x)$ will be all negative real numbers, i.e., $\{y \mid y < 0\}$.

Analyzing the Vertical Shift

However, we must also consider the vertical shift of the function $g(x)$ downward by 5 units. This means that the range of $g(x)$ will be shifted downward by 5 units, resulting in a range of $\{y \mid y < -5\}$.

Conclusion

In conclusion, the range of the function $g(x) = -f(x) - 5$ is $\{y \mid y < -5\}$. This is because the function $g(x)$ is a reflection of the function $f(x) = 10^x$ across the x-axis, followed by a vertical shift downward by 5 units.

Final Answer

The final answer is $\boxed{\{y \mid y < -5\}}$.

Introduction

In our previous article, we explored the concept of function transformation and how it applies to the given problem. We delved into the world of exponential functions, specifically the function $f(x) = 10^x$, and examined how the function $g(x) = -f(x) - 5$ is related to it. In this article, we'll answer some frequently asked questions related to the transformation of functions.

Q: What is the difference between a reflection and a vertical shift?

A: A reflection is a transformation that flips a function's graph across a line, while a vertical shift is a transformation that moves a function's graph up or down by a certain number of units.

Q: How does the reflection of a function affect its range?

A: The reflection of a function across the x-axis will result in a range that is the negative of the original range.

Q: How does the vertical shift of a function affect its range?

A: The vertical shift of a function will result in a range that is shifted up or down by a certain number of units.

Q: What is the range of the function $g(x) = -f(x) - 5$?

A: The range of the function $g(x) = -f(x) - 5$ is $\{y \mid y < -5\}$.

Q: How can I determine the range of a function after it has been transformed?

A: To determine the range of a function after it has been transformed, you need to analyze the type of transformation that has been applied. If the function has been reflected across the x-axis, you need to consider the negative of the original range. If the function has been shifted vertically, you need to consider the shift in the range.

Q: Can a function have a range that is a combination of positive and negative values?

A: No, a function cannot have a range that is a combination of positive and negative values. The range of a function is a set of all possible output values it can produce for the given input values, and it must be a single set of values.

Q: How can I visualize the transformation of a function?

A: You can visualize the transformation of a function by graphing the original function and the transformed function on the same coordinate plane. This will allow you to see the effect of the transformation on the function's graph.

Q: What is the importance of understanding function transformation?

A: Understanding function transformation is important because it allows you to analyze and interpret the behavior of functions in different contexts. It also helps you to identify patterns and relationships between functions.

Conclusion

In conclusion, understanding function transformation is a crucial concept in mathematics. By analyzing the type of transformation that has been applied to a function, you can determine the range of the function and visualize its behavior. We hope that this article has helped you to better understand the concept of function transformation and its applications.

Final Answer

The final answer is $\boxed{\{y \mid y < -5\}}$.