Find The Inverse Of The Given Function. $ F(x) = -\frac{1}{2} \sqrt{x+3}, \ X \geq -3 $ F^{-1}(x) = \ \square \ X^2 \ \square , \ \text{for} \ X \leq \ \square $

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Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function f(x), the inverse function f^{-1}(x) is a function that undoes the action of f(x). In other words, if f(x) maps an input x to an output y, then f^{-1}(x) maps the output y back to the input x. In this article, we will focus on finding the inverse of a given function f(x) = -\frac{1}{2} \sqrt{x+3}, x \geq -3.

Understanding the Given Function

Before we proceed to find the inverse of the given function, let's first understand the function itself. The function f(x) = -\frac{1}{2} \sqrt{x+3} is a square root function that takes an input x and returns an output y. The function is defined for x \geq -3, which means that the input x must be greater than or equal to -3.

Step 1: Interchange x and y

To find the inverse of the function, we need to interchange the x and y variables. This means that we will replace x with y and y with x. So, the function becomes:

x = -\frac{1}{2} \sqrt{y+3}

Step 2: Square Both Sides

Next, we need to square both sides of the equation to eliminate the square root. This will give us:

x^2 = \frac{1}{4} (y+3)

Step 3: Multiply Both Sides by 4

To simplify the equation, we can multiply both sides by 4. This will give us:

4x^2 = y+3

Step 4: Subtract 3 from Both Sides

Next, we need to subtract 3 from both sides of the equation to isolate y. This will give us:

4x^2 - 3 = y

Step 5: Write the Inverse Function

Now that we have isolated y, we can write the inverse function f^{-1}(x) as:

f^{-1}(x) = 4x^2 - 3

Domain and Range of the Inverse Function

The domain of the inverse function f^{-1}(x) is the set of all possible input values for x. Since the original function f(x) is defined for x \geq -3, the domain of the inverse function f^{-1}(x) is also x \leq 0.

Conclusion

In this article, we have found the inverse of the given function f(x) = -\frac{1}{2} \sqrt{x+3}, x \geq -3. The inverse function f^{-1}(x) is given by f^{-1}(x) = 4x^2 - 3, x \leq 0. We have also discussed the domain and range of the inverse function.

Example Problems

  1. Find the inverse of the function f(x) = 2 \sqrt{x-1}, x \geq 1.
  2. Find the inverse of the function f(x) = -3 \sqrt{x+2}, x \geq -2.

Solutions

  1. To find the inverse of the function f(x) = 2 \sqrt{x-1}, x \geq 1, we can follow the same steps as before. Interchanging x and y, we get:

x = 2 \sqrt{y-1}

Squaring both sides, we get:

x^2 = 4(y-1)

Multiplying both sides by 4, we get:

4x^2 = 4y - 4

Adding 4 to both sides, we get:

4x^2 + 4 = 4y

Dividing both sides by 4, we get:

x^2 + 1 = y

So, the inverse function f^{-1}(x) is given by f^{-1}(x) = x^2 + 1, x \leq 0.

  1. To find the inverse of the function f(x) = -3 \sqrt{x+2}, x \geq -2, we can follow the same steps as before. Interchanging x and y, we get:

x = -3 \sqrt{y+2}

Squaring both sides, we get:

x^2 = 9(y+2)

Multiplying both sides by 1/9, we get:

x^2/9 = y+2

Subtracting 2 from both sides, we get:

x^2/9 - 2 = y

So, the inverse function f^{-1}(x) is given by f^{-1}(x) = x^2/9 - 2, x \leq 0.

Tips and Tricks

  1. When finding the inverse of a function, make sure to interchange the x and y variables correctly.
  2. When squaring both sides of an equation, make sure to simplify the equation correctly.
  3. When multiplying both sides of an equation by a constant, make sure to simplify the equation correctly.

Conclusion

Q: What is an inverse function?

A: An inverse function is a function that undoes the action of the original function. In other words, if f(x) maps an input x to an output y, then f^{-1}(x) maps the output y back to the input x.

Q: Why do we need to find the inverse of a function?

A: We need to find the inverse of a function to solve equations, to graph functions, and to understand the relationship between two functions.

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

A: To find the inverse of a function, we need to follow these steps:

  1. Interchange the x and y variables.
  2. Square both sides of the equation to eliminate the square root.
  3. Multiply both sides of the equation by a constant to simplify the equation.
  4. Add or subtract a constant to isolate y.

Q: What is the domain and range of the inverse function?

A: The domain of the inverse function is the set of all possible input values for x. The range of the inverse function is the set of all possible output values for y.

Q: Can we find the inverse of any function?

A: No, we cannot find the inverse of any function. Some functions do not have an inverse, such as functions that are not one-to-one.

Q: How do we know if a function has an inverse?

A: We can check if a function has an inverse by checking if the function is one-to-one. A function is one-to-one if it passes the horizontal line test.

Q: What is the horizontal line test?

A: The horizontal line test is a test to see if a function is one-to-one. To pass the horizontal line test, a function must have no two points with the same y-coordinate.

Q: Can we find the inverse of a function that is not one-to-one?

A: No, we cannot find the inverse of a function that is not one-to-one. A function that is not one-to-one does not have an inverse.

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

A: To graph the inverse of a function, we need to reflect the graph of the original function across the line y = x.

Q: Can we find the inverse of a function that is a quadratic function?

A: Yes, we can find the inverse of a quadratic function. To find the inverse of a quadratic function, we need to follow the same steps as before.

Q: Can we find the inverse of a function that is a polynomial function?

A: Yes, we can find the inverse of a polynomial function. To find the inverse of a polynomial function, we need to follow the same steps as before.

Q: Can we find the inverse of a function that is a trigonometric function?

A: Yes, we can find the inverse of a trigonometric function. To find the inverse of a trigonometric function, we need to follow the same steps as before.

Q: Can we find the inverse of a function that is a rational function?

A: Yes, we can find the inverse of a rational function. To find the inverse of a rational function, we need to follow the same steps as before.

Q: Can we find the inverse of a function that is a logarithmic function?

A: Yes, we can find the inverse of a logarithmic function. To find the inverse of a logarithmic function, we need to follow the same steps as before.

Q: Can we find the inverse of a function that is an exponential function?

A: Yes, we can find the inverse of an exponential function. To find the inverse of an exponential function, we need to follow the same steps as before.

Conclusion

In this article, we have answered some common questions about inverse functions. We have discussed the definition of an inverse function, the steps to find the inverse of a function, and the domain and range of the inverse function. We have also discussed the horizontal line test and how to graph the inverse of a function.