Use F ( X ) = 1 2 X F(x)=\frac{1}{2} X F ( X ) = 2 1 ​ X And F − 1 ( X ) = 2 X F^{-1}(x)=2 X F − 1 ( X ) = 2 X To Solve The Problems.1. F ( 2 ) = 1 F(2)=1 F ( 2 ) = 1 2. F − 1 ( 1 ) = 2 F^{-1}(1)=2 F − 1 ( 1 ) = 2 3. F − 1 ( F ( 2 ) ) = 2 F^{-1}(f(2))=2 F − 1 ( F ( 2 )) = 2 Complete: $\begin{array}{l} f^{-1}(-2)=-4 \ f(-4)=-2 \

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Inverse Functions: Solving Problems with f(x)=12xf(x)=\frac{1}{2} x and f−1(x)=2xf^{-1}(x)=2 x

Understanding Inverse Functions

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. They play a crucial role in solving equations, graphing functions, and analyzing mathematical relationships. In this article, we will explore how to use the given functions f(x)=12xf(x)=\frac{1}{2} x and f−1(x)=2xf^{-1}(x)=2 x to solve three problems. We will also delve into the concept of inverse functions, their properties, and how they are used in mathematics.

Properties of Inverse Functions

Before we dive into the problems, let's review some essential properties of inverse functions:

  • Definition: An inverse function is a function that reverses the operation of the original function. In other words, if f(x)f(x) is a function, then its inverse function f−1(x)f^{-1}(x) satisfies the condition f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x.
  • One-to-One Correspondence: An inverse function exists only if the original function is one-to-one, meaning that each output value corresponds to exactly one input value.
  • Symmetry: The graph of an inverse function is symmetric to the graph of the original function with respect to the line y=xy = x.

Problem 1: f(2)=1f(2)=1

To solve this problem, we need to find the value of xx such that f(x)=1f(x) = 1. Since we are given the function f(x)=12xf(x) = \frac{1}{2} x, we can set up the equation 12x=1\frac{1}{2} x = 1 and solve for xx.

12x=1\frac{1}{2} x = 1

Multiplying both sides by 2, we get:

x=2x = 2

Therefore, the solution to the problem is x=2x = 2.

Problem 2: f−1(1)=2f^{-1}(1)=2

This problem involves the inverse function f−1(x)=2xf^{-1}(x) = 2 x. We need to find the value of xx such that f−1(x)=2f^{-1}(x) = 2. Since we are given the inverse function, we can set up the equation 2x=22 x = 2 and solve for xx.

2x=22 x = 2

Dividing both sides by 2, we get:

x=1x = 1

Therefore, the solution to the problem is x=1x = 1.

Problem 3: f−1(f(2))=2f^{-1}(f(2))=2

This problem involves both the original function f(x)=12xf(x) = \frac{1}{2} x and its inverse function f−1(x)=2xf^{-1}(x) = 2 x. We need to find the value of f−1(f(2))f^{-1}(f(2)). First, we need to find the value of f(2)f(2).

f(2)=12⋅2=1f(2) = \frac{1}{2} \cdot 2 = 1

Now, we need to find the value of f−1(1)f^{-1}(1). Since we are given the inverse function, we can set up the equation f−1(1)=2f^{-1}(1) = 2 and solve for f−1(1)f^{-1}(1).

f−1(1)=2f^{-1}(1) = 2

Therefore, the solution to the problem is f−1(f(2))=2f^{-1}(f(2)) = 2.

Additional Problems

In addition to the three problems above, we are given two more problems to solve:

  • f−1(−2)=−4f^{-1}(-2)=-4
  • f(−4)=−2f(-4)=-2

To solve these problems, we can use the same approach as before. For the first problem, we need to find the value of xx such that f−1(x)=−4f^{-1}(x) = -4. Since we are given the inverse function, we can set up the equation 2x=−42 x = -4 and solve for xx.

2x=−42 x = -4

Dividing both sides by 2, we get:

x=−2x = -2

Therefore, the solution to the problem is x=−2x = -2.

For the second problem, we need to find the value of xx such that f(x)=−2f(x) = -2. Since we are given the function, we can set up the equation 12x=−2\frac{1}{2} x = -2 and solve for xx.

12x=−2\frac{1}{2} x = -2

Multiplying both sides by 2, we get:

x=−4x = -4

Therefore, the solution to the problem is x=−4x = -4.

Conclusion

In this article, we have explored how to use the given functions f(x)=12xf(x) = \frac{1}{2} x and f−1(x)=2xf^{-1}(x) = 2 x to solve three problems. We have also reviewed some essential properties of inverse functions, including their definition, one-to-one correspondence, and symmetry. Additionally, we have solved two more problems using the same approach. By understanding and applying inverse functions, we can solve a wide range of mathematical problems and gain a deeper insight into the underlying mathematical relationships.

References

Understanding Inverse Functions

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. They play a crucial role in solving equations, graphing functions, and analyzing mathematical relationships. In this article, we will explore some frequently asked questions about inverse functions and provide detailed answers to help you better understand this concept.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of the original function. In other words, if f(x)f(x) is a function, then its inverse function f−1(x)f^{-1}(x) satisfies the condition 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 follow these steps:

  1. Replace f(x)f(x) with yy.
  2. Swap xx and yy.
  3. Solve for yy.

For example, if we have the function f(x)=2x+3f(x) = 2x + 3, we can find its inverse by following these steps:

  1. Replace f(x)f(x) with yy: y=2x+3y = 2x + 3
  2. Swap xx and yy: x=2y+3x = 2y + 3
  3. Solve for yy: y=x−32y = \frac{x - 3}{2}

Therefore, the inverse of the function f(x)=2x+3f(x) = 2x + 3 is f−1(x)=x−32f^{-1}(x) = \frac{x - 3}{2}.

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

A: A function and its inverse are two different functions that are related to each other. The function f(x)f(x) and its inverse f−1(x)f^{-1}(x) satisfy the condition f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x. This means that if we apply the function and its inverse to a value, we will get the original value back.

Q: How do I graph an inverse function?

A: To graph an inverse function, you need to follow these steps:

  1. Graph the original function.
  2. Reflect the graph of the original function across the line y=xy = x.

For example, if we have the function f(x)=x2f(x) = x^2, we can graph its inverse by reflecting the graph of f(x)=x2f(x) = x^2 across the line y=xy = x.

Q: What are some common applications of inverse functions?

A: Inverse functions have many applications in mathematics, science, and engineering. Some common applications include:

  • Solving equations: Inverse functions can be used to solve equations by reversing the operation of the original function.
  • Graphing functions: Inverse functions can be used to graph functions by reflecting the graph of the original function across the line y=xy = x.
  • Analyzing mathematical relationships: Inverse functions can be used to analyze mathematical relationships by reversing the operation of the original function.

Q: What are some common mistakes to avoid when working with inverse functions?

A: When working with inverse functions, there are several common mistakes to avoid:

  • Confusing the function and its inverse: Make sure to distinguish between the function and its inverse.
  • Not checking for one-to-one correspondence: Make sure that the function is one-to-one before finding its inverse.
  • Not checking for symmetry: Make sure that the graph of the function is symmetric to the line y=xy = x before graphing its inverse.

Conclusion

Inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving equations, graphing functions, and analyzing mathematical relationships. By following the steps outlined in this article, you can better understand inverse functions and apply them to a wide range of mathematical problems.

References