Which Function Has The Same Maximum Value As F ( X ) = − ∣ X + 3 ∣ − 2 F(x) = -|x+3| - 2 F ( X ) = − ∣ X + 3∣ − 2 ?A. F ( X ) = X + 3 − 3 F(x) = \sqrt{x+3} - 3 F ( X ) = X + 3 ​ − 3 B. F ( X ) = ( X + 3 ) 2 − 2 F(x) = (x+3)^2 - 2 F ( X ) = ( X + 3 ) 2 − 2 C. F ( X ) = − X + 6 − 2 F(x) = -\sqrt{x+6} - 2 F ( X ) = − X + 6 ​ − 2 D. F ( X ) = − ( X − 6 ) 2 − 3 F(x) = -(x-6)^2 - 3 F ( X ) = − ( X − 6 ) 2 − 3

by ADMIN 407 views

Which Function Has the Same Maximum Value as f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

Understanding the Problem

To determine which function has the same maximum value as f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, we need to analyze the given function and understand its behavior. The function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 involves the absolute value of x+3x+3, which means its graph will have a V-shape with its vertex at x=−3x = -3. The negative sign in front of the absolute value function causes the graph to open downwards, resulting in a maximum value at the vertex.

Analyzing the Given Function

The given function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 can be broken down into two parts:

  1. −∣x+3∣-|x+3|: This part of the function represents the absolute value of x+3x+3 with a negative sign. The absolute value function ∣x+3∣|x+3| has a minimum value of 0 at x=−3x = -3, and the negative sign in front of it causes the graph to open downwards.
  2. −2-2: This is a constant term that shifts the graph of the function downwards by 2 units.

Finding the Maximum Value

To find the maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, we need to evaluate the function at the vertex x=−3x = -3. Substituting x=−3x = -3 into the function, we get:

f(−3)=−∣−3+3∣−2=−∣0∣−2=−0−2=−2f(-3) = -|-3+3| - 2 = -|0| - 2 = -0 - 2 = -2

Therefore, the maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 is -2.

Comparing with the Options

Now, let's compare the maximum value of the given function with the options provided:

A. f(x)=x+3−3f(x) = \sqrt{x+3} - 3 B. f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 C. f(x)=−x+6−2f(x) = -\sqrt{x+6} - 2 D. f(x)=−(x−6)2−3f(x) = -(x-6)^2 - 3

We need to find which of these functions has the same maximum value as f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, which is -2.

Analyzing Option A

Option A is f(x)=x+3−3f(x) = \sqrt{x+3} - 3. To find its maximum value, we need to evaluate the function at the vertex x=−3x = -3. Substituting x=−3x = -3 into the function, we get:

f(−3)=−3+3−3=0−3=0−3=−3f(-3) = \sqrt{-3+3} - 3 = \sqrt{0} - 3 = 0 - 3 = -3

Therefore, the maximum value of option A is -3, which is not the same as the maximum value of the given function.

Analyzing Option B

Option B is f(x)=(x+3)2−2f(x) = (x+3)^2 - 2. To find its maximum value, we need to evaluate the function at the vertex x=−3x = -3. Substituting x=−3x = -3 into the function, we get:

f(−3)=(−3+3)2−2=02−2=0−2=−2f(-3) = (-3+3)^2 - 2 = 0^2 - 2 = 0 - 2 = -2

Therefore, the maximum value of option B is -2, which is the same as the maximum value of the given function.

Analyzing Option C

Option C is f(x)=−x+6−2f(x) = -\sqrt{x+6} - 2. To find its maximum value, we need to evaluate the function at the vertex x=−6x = -6. Substituting x=−6x = -6 into the function, we get:

f(−6)=−−6+6−2=−0−2=0−2=−2f(-6) = -\sqrt{-6+6} - 2 = -\sqrt{0} - 2 = 0 - 2 = -2

Therefore, the maximum value of option C is -2, which is the same as the maximum value of the given function.

Analyzing Option D

Option D is f(x)=−(x−6)2−3f(x) = -(x-6)^2 - 3. To find its maximum value, we need to evaluate the function at the vertex x=6x = 6. Substituting x=6x = 6 into the function, we get:

f(6)=−(6−6)2−3=−(0)2−3=0−3=−3f(6) = -(6-6)^2 - 3 = -(0)^2 - 3 = 0 - 3 = -3

Therefore, the maximum value of option D is -3, which is not the same as the maximum value of the given function.

Conclusion

Based on the analysis, we can conclude that options B and C have the same maximum value as the given function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2. However, since the question asks for a single function, we need to choose one of the options that have the same maximum value.

Therefore, the correct answer is:

B. f(x)=(x+3)2−2f(x) = (x+3)^2 - 2

This function has the same maximum value as the given function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, which is -2.
Q&A: Understanding the Function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2

Q: What is the vertex of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: The vertex of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 is at x=−3x = -3. This is because the absolute value function ∣x+3∣|x+3| has a minimum value of 0 at x=−3x = -3, and the negative sign in front of it causes the graph to open downwards.

Q: What is the maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: The maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 is -2. This is because the function has a V-shape with its vertex at x=−3x = -3, and the negative sign in front of the absolute value function causes the graph to open downwards.

Q: How do you find the maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: To find the maximum value of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, you need to evaluate the function at the vertex x=−3x = -3. Substituting x=−3x = -3 into the function, you get:

f(−3)=−∣−3+3∣−2=−∣0∣−2=−0−2=−2f(-3) = -|-3+3| - 2 = -|0| - 2 = -0 - 2 = -2

Q: What is the difference between the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 and the function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2?

A: The main difference between the two functions is the presence of the absolute value function in the first function. The function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 has a V-shape with its vertex at x=−3x = -3, while the function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 has a parabolic shape with its vertex at x=−3x = -3.

Q: Why is the function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 a better choice than the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: The function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 is a better choice than the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 because it has a more straightforward and easier-to-understand form. The function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 is a quadratic function, which means it can be easily analyzed and understood using algebraic techniques.

Q: Can you provide more examples of functions that have the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: Yes, here are a few more examples of functions that have the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2:

  • f(x)=−(x−6)2−3f(x) = -(x-6)^2 - 3
  • f(x)=−x+6−2f(x) = -\sqrt{x+6} - 2
  • f(x)=−(x+6)2−4f(x) = -(x+6)^2 - 4

These functions all have the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, which is -2.

Q: How do you determine which function has the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2?

A: To determine which function has the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, you need to evaluate each function at the vertex x=−3x = -3 and compare the results. If the function has the same maximum value as the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2, then it is a valid choice.

Q: What is the significance of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 in real-world applications?

A: The function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2 has several real-world applications, including:

  • Modeling the behavior of a physical system that has a V-shape with its vertex at x=−3x = -3.
  • Analyzing the behavior of a system that has a maximum value at x=−3x = -3.
  • Developing algorithms for solving optimization problems that involve the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2.

These are just a few examples of the many real-world applications of the function f(x)=−∣x+3∣−2f(x) = -|x+3| - 2.