What Is The Range Of The Function $f(x)=-4|x+1|-5$?A. $(-\infty,-5]$B. $ [ − 5 , ∞ ) [-5, \infty) [ − 5 , ∞ ) [/tex]C. $\left[-4, \infty\right)$D. $(-\infty,-4]$
Introduction
When dealing with functions, understanding the range is crucial in determining the possible output values. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we will explore the range of the function $f(x)=-4|x+1|-5$ and determine the correct answer among the given options.
Understanding Absolute Value Functions
Before diving into the specific function, let's briefly discuss absolute value functions. An absolute value function is a function that takes the absolute value of a real number as its input. The general form of an absolute value function is $f(x)=|x-a|+b$, where $a$ and $b$ are constants. The absolute value function has two cases: when $x\geq a$ and when $x< a$.
Case 1: When $x\geq -1$
When $x\geq -1$, the absolute value function $|x+1|$ simplifies to $x+1$. Therefore, the function $f(x)=-4|x+1|-5$ becomes $f(x)=-4(x+1)-5=-4x-9$.
Case 2: When $x<-1$
When $x<-1$, the absolute value function $|x+1|$ simplifies to $-(x+1)=-x-1$. Therefore, the function $f(x)=-4|x+1|-5$ becomes $f(x)=-4(-x-1)-5=4x-1$.
Combining the Cases
To find the range of the function, we need to consider both cases. When $x\geq -1$, the function is $f(x)=-4x-9$, and when $x<-1$, the function is $f(x)=4x-1$.
Finding the Range
To find the range, we need to determine the minimum and maximum values of the function. Let's analyze the two cases separately.
Case 1: When $x\geq -1$
In this case, the function is $f(x)=-4x-9$. To find the minimum value, we can take the derivative of the function and set it equal to zero. The derivative is $f'(x)=-4$, which is a constant. Since the derivative is constant, the function is a linear function, and its minimum value occurs at the endpoint $x=-1$. Plugging in $x=-1$, we get $f(-1)=-4(-1)-9=-5$.
Case 2: When $x<-1$
In this case, the function is $f(x)=4x-1$. To find the minimum value, we can take the derivative of the function and set it equal to zero. The derivative is $f'(x)=4$, which is a constant. Since the derivative is constant, the function is a linear function, and its minimum value occurs at the endpoint $x=-\infty$. As $x$ approaches $-\infty$, the function approaches $-\infty$.
Combining the Results
From the analysis of both cases, we can see that the minimum value of the function is $-5$, which occurs at $x=-1$. The function approaches $-\infty$ as $x$ approaches $-\infty$. Therefore, the range of the function is $(-\infty,-5]$.
Conclusion
In conclusion, the range of the function $f(x)=-4|x+1|-5$ is $(-\infty,-5]$. This is the correct answer among the given options.
Final Answer
The final answer is $\boxed{A}$.
Discussion
The range of a function is an essential concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we explored the range of the function $f(x)=-4|x+1|-5$ and determined the correct answer among the given options. We analyzed the function in two cases: when $x\geq -1$ and when $x<-1$. By finding the minimum and maximum values of the function in each case, we determined that the range of the function is $(-\infty,-5]$. This is an important concept to understand in mathematics, and we hope this article has provided a clear explanation of the range of the function $f(x)=-4|x+1|-5$.
Introduction
In our previous article, we explored the range of the function $f(x)=-4|x+1|-5$. We analyzed the function in two cases: when $x\geq -1$ and when $x<-1$. By finding the minimum and maximum values of the function in each case, we determined that the range of the function is $(-\infty,-5]$. In this article, we will answer some frequently asked questions about the range of the function.
Q1: What is the range of the function $f(x)=-4|x+1|-5$?
A1: The range of the function $f(x)=-4|x+1|-5$ is $(-\infty,-5]$.
Q2: Why is the range of the function $(-\infty,-5]$?
A2: The range of the function is $(-\infty,-5]$ because the function approaches $-\infty$ as $x$ approaches $-\infty$, and the minimum value of the function is $-5$, which occurs at $x=-1$.
Q3: What happens to the function as $x$ approaches $-\infty$?
A3: As $x$ approaches $-\infty$, the function approaches $-\infty$.
Q4: What is the minimum value of the function?
A4: The minimum value of the function is $-5$, which occurs at $x=-1$.
Q5: How do we determine the range of the function?
A5: We determine the range of the function by analyzing the function in two cases: when $x\geq -1$ and when $x<-1$. By finding the minimum and maximum values of the function in each case, we can determine the range of the function.
Q6: What is the significance of the range of the function?
A6: The range of the function is significant because it tells us the possible output values of the function for the given input values. Understanding the range of the function is crucial in solving problems involving functions.
Q7: Can we use the range of the function to solve problems?
A7: Yes, we can use the range of the function to solve problems. By understanding the range of the function, we can determine the possible output values of the function and use this information to solve problems.
Q8: How do we use the range of the function to solve problems?
A8: We use the range of the function to solve problems by analyzing the function and determining the possible output values. We can then use this information to solve problems involving the function.
Q9: What are some common applications of the range of the function?
A9: Some common applications of the range of the function include solving problems involving functions, determining the possible output values of the function, and understanding the behavior of the function.
Q10: Why is understanding the range of the function important?
A10: Understanding the range of the function is important because it helps us to solve problems involving functions, determine the possible output values of the function, and understand the behavior of the function.
Conclusion
In conclusion, the range of the function $f(x)=-4|x+1|-5$ is $(-\infty,-5]$. We answered some frequently asked questions about the range of the function and provided explanations for each question. Understanding the range of the function is crucial in solving problems involving functions, and we hope this article has provided a clear explanation of the range of the function.
Final Answer
The final answer is $\boxed{A}$.
Discussion
The range of a function is an essential concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we answered some frequently asked questions about the range of the function $f(x)=-4|x+1|-5$ and provided explanations for each question. We hope this article has provided a clear explanation of the range of the function and has helped to clarify any confusion.