What Is The Inverse Of The Function F ( X ) = X 2 − 9 F(x) = X^2 - 9 F ( X ) = X 2 − 9 , Where X ≥ 0 X \geq 0 X ≥ 0 ?A. F − 1 ( X ) = X − 9 F^{-1}(x) = \sqrt{x-9} F − 1 ( X ) = X − 9 ​ , Where The Domain Of F − 1 ( X F^{-1}(x F − 1 ( X ] Is Limited To Real Numbers Greater Than Or Equal To 9.B. $f^{-1}(x) =

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Understanding the Concept of Inverse Functions

Inverse functions are a crucial concept in mathematics, particularly in calculus and algebra. The inverse of a function essentially reverses the operation of the original function. In other words, if we have a function f(x)f(x), its inverse is denoted as f−1(x)f^{-1}(x), and it satisfies the property that f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x. In this article, we will explore the inverse of the function f(x)=x2−9f(x) = x^2 - 9, where x≥0x \geq 0.

The Function f(x)=x2−9f(x) = x^2 - 9

The given function is f(x)=x2−9f(x) = x^2 - 9, where x≥0x \geq 0. This function represents a quadratic function that opens upwards, with its vertex at (0,−9)(0, -9). Since the domain of the function is restricted to x≥0x \geq 0, the function is only defined for non-negative values of xx.

Finding the Inverse of the Function

To find the inverse of the function f(x)=x2−9f(x) = x^2 - 9, we need to solve the equation y=x2−9y = x^2 - 9 for xx. This will give us the inverse function f−1(x)f^{-1}(x).

Step 1: Interchange xx and yy

We start by interchanging xx and yy in the equation y=x2−9y = x^2 - 9. This gives us x=y2−9x = y^2 - 9.

Step 2: Solve for yy

Next, we solve for yy by isolating it on one side of the equation. We add 9 to both sides of the equation to get x+9=y2x + 9 = y^2.

Step 3: Take the Square Root

Since we are looking for the inverse function, we need to take the square root of both sides of the equation. This gives us y=±x+9y = \pm \sqrt{x + 9}.

Step 4: Restrict the Domain

Since the original function is restricted to x≥0x \geq 0, we need to restrict the domain of the inverse function accordingly. This means that we only consider the positive square root, which gives us y=x+9y = \sqrt{x + 9}.

The Inverse Function f−1(x)f^{-1}(x)

The inverse function f−1(x)f^{-1}(x) is given by f−1(x)=x+9f^{-1}(x) = \sqrt{x + 9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9.

Comparison with the Given Options

Let's compare our result with the given options:

A. f−1(x)=x−9f^{-1}(x) = \sqrt{x-9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to 9.

B. f−1(x)=x+9f^{-1}(x) = \sqrt{x+9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9.

Our result matches option B, which is f−1(x)=x+9f^{-1}(x) = \sqrt{x+9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9.

Conclusion

In conclusion, the inverse of the function f(x)=x2−9f(x) = x^2 - 9, where x≥0x \geq 0, is given by f−1(x)=x+9f^{-1}(x) = \sqrt{x + 9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9. This result is consistent with option B, which is the correct answer.

Frequently Asked Questions

Q: What is the inverse of the function f(x)=x2−9f(x) = x^2 - 9?

A: The inverse of the function f(x)=x2−9f(x) = x^2 - 9 is given by f−1(x)=x+9f^{-1}(x) = \sqrt{x + 9}.

Q: What is the domain of the inverse function f−1(x)f^{-1}(x)?

A: The domain of the inverse function f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to solve the equation y=f(x)y = f(x) for xx. This will give you the inverse function f−1(x)f^{-1}(x).

Final Thoughts

Inverse functions are a crucial concept in mathematics, and understanding how to find them is essential for solving problems in calculus and algebra. In this article, we explored the inverse of the function f(x)=x2−9f(x) = x^2 - 9, where x≥0x \geq 0. We found that the inverse function is given by f−1(x)=x+9f^{-1}(x) = \sqrt{x + 9}, where the domain of f−1(x)f^{-1}(x) is limited to real numbers greater than or equal to -9. We hope that this article has provided you with a better understanding of inverse functions and how to find them.

Understanding Inverse Functions

Inverse functions are a crucial concept in mathematics, particularly in calculus and algebra. The inverse of a function essentially reverses the operation of the original function. In other words, if we have a function f(x)f(x), its inverse is denoted as f−1(x)f^{-1}(x), and it satisfies the property that f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x. In this article, we will answer some frequently asked questions about inverse functions.

Q&A Session

Q: What is the inverse of a function?

A: The inverse of a function is a function that reverses the operation of the original function. In other words, if we have a function f(x)f(x), its inverse is denoted as f−1(x)f^{-1}(x), and it satisfies the property that f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to solve the equation y=f(x)y = f(x) for xx. This will give you the inverse function f−1(x)f^{-1}(x).

Q: What is the domain of the inverse function?

A: The domain of the inverse function is the set of all possible input values for the inverse function. In other words, it is the set of all possible values of xx for which the inverse function is defined.

Q: How do I determine the domain of the inverse function?

A: To determine the domain of the inverse function, you need to consider the original function and its restrictions. If the original function is restricted to a certain domain, then the inverse function will also be restricted to the same domain.

Q: What is the range of the inverse function?

A: The range of the inverse function is the set of all possible output values for the inverse function. In other words, it is the set of all possible values of f−1(x)f^{-1}(x).

Q: How do I determine the range of the inverse function?

A: To determine the range of the inverse function, you need to consider the original function and its restrictions. If the original function is restricted to a certain range, then the inverse function will also be restricted to the same range.

Q: What is the difference between the domain and range of a function and its inverse?

A: The domain of a function is the set of all possible input values for the function, while the range of a function is the set of all possible output values for the function. The domain and range of a function and its inverse are related but not identical. The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.

Q: How do I graph the inverse of a function?

A: To graph the inverse of a function, you need to reflect the original function about the line y=xy = x. This will give you the graph of the inverse function.

Q: What is the importance of inverse functions in real-world applications?

A: Inverse functions have many real-world applications, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as population growth, chemical reactions, and financial transactions.

Conclusion

In conclusion, inverse functions are a crucial concept in mathematics, and understanding how to find them is essential for solving problems in calculus and algebra. In this article, we answered some frequently asked questions about inverse functions, including how to find the inverse of a function, the domain and range of the inverse function, and how to graph the inverse of a function. We hope that this article has provided you with a better understanding of inverse functions and their applications.

Final Thoughts

Inverse functions are a powerful tool in mathematics, and understanding how to find them is essential for solving problems in calculus and algebra. In this article, we explored the concept of inverse functions and answered some frequently asked questions about them. We hope that this article has provided you with a better understanding of inverse functions and their applications.

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