Type The Correct Answer In Each Box. Use Numerals Instead Of Words.What Is The Inverse Of This Function?$[ \begin{array}{l} f(x) = -\frac{1}{2} \sqrt{x+3}, , X \geq -3 \ f^{-1}(x) = \square X^2 - \square, , \text{for } X \leq
Inverse Functions: Understanding the Concept and Calculating the Inverse of a Given Function
What is an Inverse Function?
An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^{-1}(x) maps the output f(x) back to the original input x. Inverse functions are denoted by the notation f^{-1}(x) and are used to "undo" the operation of the original function.
The Given Function
The given function is f(x) = -\frac{1}{2} \sqrt{x+3}, where x \geq -3. This function takes an input x, adds 3 to it, takes the square root of the result, and then multiplies it by -\frac{1}{2}.
The Inverse Function
To find the inverse function f^{-1}(x), we need to reverse the operation of the original function. This means that we need to undo the multiplication by -\frac{1}{2}, the taking of the square root, and the addition of 3.
Step 1: Undo the Multiplication by -\frac{1}{2}
To undo the multiplication by -\frac{1}{2}, we need to multiply both sides of the equation by -2. This gives us:
f^{-1}(x) = -2 \cdot f(x) = -2 \cdot \left(-\frac{1}{2} \sqrt{x+3}\right) = \sqrt{x+3}
Step 2: Undo the Taking of the Square Root
To undo the taking of the square root, we need to square both sides of the equation. This gives us:
f^{-1}(x) = \left(\sqrt{x+3}\right)^2 = x+3
Step 3: Undo the Addition of 3
To undo the addition of 3, we need to subtract 3 from both sides of the equation. This gives us:
f^{-1}(x) = (x+3) - 3 = x
The Final Answer
Therefore, the inverse function f^{-1}(x) is:
f^{-1}(x) = x
However, we need to consider the domain and range of the original function. The original function is defined for x \geq -3, so the inverse function is also defined for x \geq -3. Therefore, the final answer is:
f^{-1}(x) = \boxed{x^2 - 3}, , \text{for } x \leq \boxed{-\frac{1}{4}}
Conclusion
In this article, we have discussed the concept of inverse functions and calculated the inverse of a given function. We have shown that the inverse function is f^{-1}(x) = x, but we have also considered the domain and range of the original function. The final answer is f^{-1}(x) = x^2 - 3, for x \leq -\frac{1}{4}.
Inverse Functions: A Q&A Guide
Frequently Asked Questions
Q: What is an inverse function?
A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^{-1}(x) maps the output f(x) back to the original input x.
Q: Why do we need inverse functions?
A: Inverse functions are used to "undo" the operation of the original function. This is useful in many areas of mathematics, science, and engineering, such as solving equations, finding the roots of a function, and modeling real-world phenomena.
Q: How do we find the inverse of a function?
A: To find the inverse of a function, we need to reverse the operation of the original function. This involves undoing any operations such as multiplication, division, addition, or subtraction, and then solving for the input variable.
Q: What are some common mistakes to avoid when finding the inverse of a function?
A: Some common mistakes to avoid when finding the inverse of a function include:
- Not reversing the operation of the original function
- Not solving for the input variable
- Not considering the domain and range of the original function
- Not checking for any restrictions on the domain and range of the inverse function
Q: How do we check if a function is one-to-one?
A: A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place. This is equivalent to saying that the function is either strictly increasing or strictly decreasing.
Q: What is the difference between a one-to-one function and an invertible function?
A: A one-to-one function is a function that passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place. An invertible function is a function that has an inverse function, meaning that it is one-to-one and onto.
Q: How do we find the inverse of a composite function?
A: To find the inverse of a composite function, we need to use the chain rule of differentiation. This involves differentiating the outer function with respect to the inner function, and then solving for the input variable.
Q: What are some real-world applications of inverse functions?
A: Inverse functions have many real-world applications, such as:
- Modeling population growth and decline
- Analyzing the behavior of electrical circuits
- Solving optimization problems in economics and finance
- Modeling the motion of objects in physics and engineering
Conclusion
In this article, we have discussed some frequently asked questions about inverse functions. We have covered topics such as the definition of an inverse function, how to find the inverse of a function, and some common mistakes to avoid. We have also discussed the difference between a one-to-one function and an invertible function, and how to find the inverse of a composite function. Finally, we have highlighted some real-world applications of inverse functions.