## Unlocking the Mystery: Finding the Missing Value in the Inverse Function

**Question:** How do you find the missing value in the inverse function, *h(x) = 2x - ?*, given the original function *f(x) = (1/2)x + 10*?

**Answer:** The missing value is **20**.

**Why is this important?** Understanding inverse functions is crucial in various fields like mathematics, physics, and computer science. It helps us reverse the process of a function, enabling us to find the input that produces a given output.

**Our analysis:** To find the missing value, we need to understand the relationship between a function and its inverse. If *f(x)* and *h(x)* are inverse functions, then:

**f(h(x)) = x**and**h(f(x)) = x**

We can use this property to determine the missing value.

**Key takeaways:**

Key Aspect | Description |
---|---|

Inverse Functions |
Two functions are inverses if they "undo" each other. |

Composition of Inverse Functions |
The composition of a function and its inverse always results in the original input (x). |

**Let's delve into the details:**

### Finding the Missing Value

**Start with the original function:***f(x) = (1/2)x + 10***Substitute the inverse function***h(x)*into the original function:*f(h(x)) = (1/2)(2x - ?) + 10***Simplify:***f(h(x)) = x - ?/2 + 10***We know f(h(x)) = x, so set the simplified expression equal to x:***x - ?/2 + 10 = x***Solve for the missing value:**- Subtract
*x*from both sides: -?/2 + 10 = 0 - Subtract
*10*from both sides: -?/2 = -10 - Multiply both sides by
*-2*: ? = 20

- Subtract

Therefore, the missing value in the inverse function *h(x) = 2x - ?* is **20**.

### Verification

We can verify our answer by checking if *h(f(x)) = x*:

**Substitute***f(x)*into the inverse function:*h(f(x)) = 2((1/2)x + 10) - 20***Simplify:***h(f(x)) = x + 20 - 20 = x*

This confirms that *h(x) = 2x - 20* is indeed the inverse of *f(x) = (1/2)x + 10*.

### FAQ

**Q: What is the significance of inverse functions?**

**A:** Inverse functions help us reverse the process of a function. They allow us to find the input that produces a given output. This is valuable in many applications, including:

**Solving equations:**Finding the solution to an equation often involves finding the inverse of a function.**Cryptography:**Inverse functions are used in encryption and decryption techniques to secure data.**Computer programming:**Inverse functions are used in various algorithms, particularly in data manipulation and transformation.

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

**A:** To find the inverse of a function *f(x)*:

**Replace***f(x)*with*y*.**Swap***x*and*y*.**Solve for***y*.**Replace***y*with*f⁻¹(x)*.

**Q: Is every function invertible?**

**A:** No, not every function has an inverse. A function must be one-to-one (meaning each input has a unique output) to have an inverse.

### Tips for Working with Inverse Functions

**Remember the relationship between a function and its inverse:***f(h(x)) = x*and*h(f(x)) = x*.**Practice finding the inverse of various functions.****Utilize online tools or calculators to verify your answers and explore different functions.****Visualize the relationship between a function and its inverse using graphs.**

### Summary

This article delved into finding the missing value in the inverse function *h(x) = 2x - ?*, given the original function *f(x) = (1/2)x + 10*. We discovered that the missing value is **20**. Understanding the concept of inverse functions is crucial in various mathematical and scientific fields, enabling us to reverse the process of a function and solve various problems.

**Moving forward:** Continue exploring the concept of inverse functions and their applications. Experiment with different functions and practice finding their inverses. This will deepen your understanding of this fundamental mathematical concept and its real-world implications.