Select All The Correct Answers.Consider The Parent Function $f(x) = E^x$ And The Transformed Function $g(x) = -f(x) - 4$. Which Features Of Function $f$ And Function $g$ Are Different?- Range- End Behavior- Domain-

Introduction

When working with functions, transformations play a crucial role in understanding their behavior and characteristics. In this article, we will explore the impact of transformations on the features of a function, specifically the parent function $f(x) = e^x$ and its transformed function $g(x) = -f(x) - 4$. We will examine the differences in the range, end behavior, and domain of these two functions.

Understanding the Parent Function

The parent function $f(x) = e^x$ is an exponential function with a base of $e$. This function has several key features that are essential to understanding its behavior:

• Domain: The domain of $f(x) = e^x$ is all real numbers, denoted as $(-\infty, \infty)$.
• Range: The range of $f(x) = e^x$ is all positive real numbers, denoted as $(0, \infty)$.
• End Behavior: As $x$ approaches negative infinity, $f(x) = e^x$ approaches 0. As $x$ approaches positive infinity, $f(x) = e^x$ approaches infinity.

Transforming the Function

The transformed function $g(x) = -f(x) - 4$ is obtained by applying three transformations to the parent function $f(x) = e^x$:

1. Reflection: The function $f(x) = e^x$ is reflected about the x-axis to obtain $-f(x)$.
2. Translation: The function $-f(x)$ is translated 4 units down to obtain $-f(x) - 4$.

Analyzing the Features of the Transformed Function

Now that we have the transformed function $g(x) = -f(x) - 4$, let's analyze its features:

• Domain: The domain of $g(x) = -f(x) - 4$ is the same as the domain of $f(x) = e^x$, which is all real numbers, denoted as $(-\infty, \infty)$.
• Range: The range of $g(x) = -f(x) - 4$ is all negative real numbers, denoted as $(-\infty, 0)$.
• End Behavior: As $x$ approaches negative infinity, $g(x) = -f(x) - 4$ approaches negative infinity. As $x$ approaches positive infinity, $g(x) = -f(x) - 4$ approaches negative infinity.

Comparing the Features of the Parent and Transformed Functions

Now that we have analyzed the features of both functions, let's compare them:

Feature	Parent Function $f(x) = e^x$	Transformed Function $g(x) = -f(x) - 4$
Domain	All real numbers, $(-\infty, \infty)$	All real numbers, $(-\infty, \infty)$
Range	All positive real numbers, $(0, \infty)$	All negative real numbers, $(-\infty, 0)$
End Behavior	Approaches 0 as $x$ approaches negative infinity, approaches infinity as $x$ approaches positive infinity	Approaches negative infinity as $x$ approaches negative infinity, approaches negative infinity as $x$ approaches positive infinity

Conclusion

In conclusion, the transformed function $g(x) = -f(x) - 4$ has different features compared to the parent function $f(x) = e^x$. The range and end behavior of the transformed function are different from the parent function, while the domain remains the same. Understanding these transformations is essential in analyzing and working with functions in mathematics.

Key Takeaways

• The domain of a function remains the same after transformations.
• The range and end behavior of a function can change after transformations.
• Reflection and translation can change the range and end behavior of a function.

Practice Problems

1. What is the domain of the function $f(x) = e^x$?
2. What is the range of the function $f(x) = e^x$?
3. What is the end behavior of the function $f(x) = e^x$?
4. What is the domain of the transformed function $g(x) = -f(x) - 4$?
5. What is the range of the transformed function $g(x) = -f(x) - 4$?
6. What is the end behavior of the transformed function $g(x) = -f(x) - 4$?

Answers

1. The domain of the function $f(x) = e^x$ is all real numbers, $(-\infty, \infty)$.
2. The range of the function $f(x) = e^x$ is all positive real numbers, $(0, \infty)$.
3. The end behavior of the function $f(x) = e^x$ is that it approaches 0 as $x$ approaches negative infinity, and approaches infinity as $x$ approaches positive infinity.
4. The domain of the transformed function $g(x) = -f(x) - 4$ is all real numbers, $(-\infty, \infty)$.
5. The range of the transformed function $g(x) = -f(x) - 4$ is all negative real numbers, $(-\infty, 0)$.
6. The end behavior of the transformed function $g(x) = -f(x) - 4$ is that it approaches negative infinity as $x$ approaches negative infinity, and approaches negative infinity as $x$ approaches positive infinity.
   Q&A: Transforming Functions =============================

Q1: What is the domain of the function $f(x) = e^x$?

A1: The domain of the function $f(x) = e^x$ is all real numbers, denoted as $(-\infty, \infty)$.

Q2: What is the range of the function $f(x) = e^x$?

A2: The range of the function $f(x) = e^x$ is all positive real numbers, denoted as $(0, \infty)$.

Q3: What is the end behavior of the function $f(x) = e^x$?

A3: The end behavior of the function $f(x) = e^x$ is that it approaches 0 as $x$ approaches negative infinity, and approaches infinity as $x$ approaches positive infinity.

Q4: What is the domain of the transformed function $g(x) = -f(x) - 4$?

A4: The domain of the transformed function $g(x) = -f(x) - 4$ is all real numbers, denoted as $(-\infty, \infty)$.

Q5: What is the range of the transformed function $g(x) = -f(x) - 4$?

A5: The range of the transformed function $g(x) = -f(x) - 4$ is all negative real numbers, denoted as $(-\infty, 0)$.

Q6: What is the end behavior of the transformed function $g(x) = -f(x) - 4$?

A6: The end behavior of the transformed function $g(x) = -f(x) - 4$ is that it approaches negative infinity as $x$ approaches negative infinity, and approaches negative infinity as $x$ approaches positive infinity.

Q7: How do transformations affect the domain of a function?

A7: Transformations do not affect the domain of a function. The domain of a function remains the same after transformations.

Q8: How do transformations affect the range of a function?

A8: Transformations can affect the range of a function. Reflection and translation can change the range of a function.

Q9: How do transformations affect the end behavior of a function?

A9: Transformations can affect the end behavior of a function. Reflection and translation can change the end behavior of a function.

Q10: What is the difference between a reflection and a translation?

A10: A reflection is a transformation that flips a function over a line, while a translation is a transformation that moves a function up or down.

Q11: How do you reflect a function over the x-axis?

A11: To reflect a function over the x-axis, you multiply the function by -1.

Q12: How do you translate a function up or down?

A12: To translate a function up or down, you add or subtract a value from the function.

Q13: What is the parent function $f(x) = e^x$?

A13: The parent function $f(x) = e^x$ is an exponential function with a base of $e$.

Q14: What is the transformed function $g(x) = -f(x) - 4$?

A14: The transformed function $g(x) = -f(x) - 4$ is obtained by reflecting the parent function $f(x) = e^x$ over the x-axis and translating it 4 units down.

Q15: What are the key takeaways from this article?

A15: The key takeaways from this article are that the domain of a function remains the same after transformations, the range and end behavior of a function can change after transformations, and reflection and translation can change the range and end behavior of a function.