Which Function Has The Same Range As F ( X ) = − 2 X − 3 + 8 F(x) = -2 \sqrt{x-3} + 8 F ( X ) = − 2 X − 3 ​ + 8 ?A. G ( X ) = X − 3 − 8 G(x) = \sqrt{x-3} - 8 G ( X ) = X − 3 ​ − 8 B. G ( X ) = X − 3 + 8 G(x) = \sqrt{x-3} + 8 G ( X ) = X − 3 ​ + 8 C. G ( X ) = − X + 3 + 8 G(x) = -\sqrt{x+3} + 8 G ( X ) = − X + 3 ​ + 8 D. G ( X ) = − X − 3 − 8 G(x) = -\sqrt{x-3} - 8 G ( X ) = − X − 3 ​ − 8

Understanding the Problem

To determine which function has the same range as $f(x) = -2 \sqrt{x-3} + 8$, we need to understand the concept of range in mathematics. The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can take.

Analyzing the Given Function

The given function is $f(x) = -2 \sqrt{x-3} + 8$. This function involves a square root term, which means it is defined only for non-negative values of the expression inside the square root. In this case, the expression inside the square root is $x-3$, which means the function is defined only for $x \geq 3$.

Determining the Range of the Function

To determine the range of the function, we need to find the minimum and maximum values it can take. Since the function involves a square root term, it is a decreasing function for $x \geq 3$. This means that as $x$ increases, the value of the function decreases.

Finding the Minimum Value

To find the minimum value of the function, we need to find the value of $x$ that makes the square root term equal to zero. This occurs when $x-3 = 0$, which gives $x = 3$. Substituting this value into the function, we get:

$f(3) = -2 \sqrt{3-3} + 8 = -2 \sqrt{0} + 8 = 8$

Finding the Maximum Value

To find the maximum value of the function, we need to find the value of $x$ that makes the square root term equal to infinity. However, since the square root term is always non-negative, it can never be equal to infinity. Therefore, the function has no maximum value.

Determining the Range

Based on the analysis above, we can conclude that the range of the function $f(x) = -2 \sqrt{x-3} + 8$ is all real numbers greater than or equal to 8.

Comparing with the Options

Now, let's compare the range of the function with the options given:

A. $g(x) = \sqrt{x-3} - 8$ B. $g(x) = \sqrt{x-3} + 8$ C. $g(x) = -\sqrt{x+3} + 8$ D. $g(x) = -\sqrt{x-3} - 8$

Analyzing Option A

Option A is $g(x) = \sqrt{x-3} - 8$. This function involves a square root term, which means it is defined only for non-negative values of the expression inside the square root. In this case, the expression inside the square root is $x-3$, which means the function is defined only for $x \geq 3$. However, the function is increasing for $x \geq 3$, which means it can take on any value greater than or equal to -8.

Analyzing Option B

Option B is $g(x) = \sqrt{x-3} + 8$. This function also involves a square root term, which means it is defined only for non-negative values of the expression inside the square root. In this case, the expression inside the square root is $x-3$, which means the function is defined only for $x \geq 3$. However, the function is increasing for $x \geq 3$, which means it can take on any value greater than or equal to 8.

Analyzing Option C

Option C is $g(x) = -\sqrt{x+3} + 8$. This function involves a square root term, which means it is defined only for non-negative values of the expression inside the square root. In this case, the expression inside the square root is $x+3$, which means the function is defined only for $x \geq -3$. However, the function is decreasing for $x \geq -3$, which means it can take on any value less than or equal to 8.

Analyzing Option D

Option D is $g(x) = -\sqrt{x-3} - 8$. This function involves a square root term, which means it is defined only for non-negative values of the expression inside the square root. In this case, the expression inside the square root is $x-3$, which means the function is defined only for $x \geq 3$. However, the function is decreasing for $x \geq 3$, which means it can take on any value less than or equal to -8.

Conclusion

Based on the analysis above, we can conclude that the function $g(x) = \sqrt{x-3} + 8$ has the same range as $f(x) = -2 \sqrt{x-3} + 8$.

The Final Answer is B.

Introduction

In our previous article, we discussed how to determine the range of a function and applied this concept to the function $f(x) = -2 \sqrt{x-3} + 8$. We also compared the range of this function with the options given and concluded that the function $g(x) = \sqrt{x-3} + 8$ has the same range as $f(x) = -2 \sqrt{x-3} + 8$.

Q&A Session

Q1: What is the range of a function?

A1: The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can take.

Q2: How do you determine the range of a function?

A2: To determine the range of a function, you need to find the minimum and maximum values it can take. This can be done by analyzing the function and its behavior.

Q3: What is the minimum value of the function $f(x) = -2 \sqrt{x-3} + 8$?

A3: The minimum value of the function $f(x) = -2 \sqrt{x-3} + 8$ is 8, which occurs when $x = 3$.

Q4: What is the maximum value of the function $f(x) = -2 \sqrt{x-3} + 8$?

A4: The function $f(x) = -2 \sqrt{x-3} + 8$ has no maximum value, as the square root term is always non-negative and can never be equal to infinity.

Q5: Which function has the same range as $f(x) = -2 \sqrt{x-3} + 8$?

A5: The function $g(x) = \sqrt{x-3} + 8$ has the same range as $f(x) = -2 \sqrt{x-3} + 8$.

Q6: What is the range of the function $g(x) = \sqrt{x-3} + 8$?

A6: The range of the function $g(x) = \sqrt{x-3} + 8$ is all real numbers greater than or equal to 8.

Q7: How do you compare the range of two functions?

A7: To compare the range of two functions, you need to analyze the behavior of each function and determine the minimum and maximum values they can take.

Q8: What is the importance of understanding the range of a function?

A8: Understanding the range of a function is important in mathematics and real-world applications, as it helps us to determine the possible output values of a function and make informed decisions.

Conclusion

In this Q&A article, we discussed the concept of range in mathematics and applied it to the function $f(x) = -2 \sqrt{x-3} + 8$. We also compared the range of this function with the options given and concluded that the function $g(x) = \sqrt{x-3} + 8$ has the same range as $f(x) = -2 \sqrt{x-3} + 8$. We hope this article has been helpful in understanding the range of functions and its importance in mathematics and real-world applications.

The Final Answer is B.