The **inverse** of a **function** is essentially a **reflection** of the original **function** across the line **( y = x )**. If I have a **function ( f(x) = 2x + 1 )**, to find its **inverse**, I need to switch the roles of **( x )** and **( y ),** and then solve for **( y )**.

This process determines a new **function**, often denoted as $f^{-1}(x) $, that will reverse the effect of the original **function**. When **f(x)** and** $f^{-1}(x)$** are composed together, the result is the input value **( x )**.

In the case of the **linear function** **( f(x) = 2x + 1 )**, finding the **inverse** involves a few **algebraic steps.** The first step is to replace** ( f(x) )** with **( y )**, resulting in the equation **( y = 2x + 1 )**. Then, I must interchange **( x )** and** ( y )** to get **( x = 2y + 1 ). **

The final step is to **solve** for **( y )**, which will give me the **formula** for the **inverse function**. Stay tuned to uncover the exact steps and the answer!

## Finding the Inverse of f(x) = 2x + 1

To find the **inverse function** of the **linear function** ( f(x) = 2x + 1 ), I first consider it as ( y = 2x + 1 ). The goal is to express ( x ) in terms of ( y ), which means I need to **solve for y**.

Here’s how I do it step by step:

I start with the original equation:

$$ y = 2x + 1$$

I then

**subtract**1 from both sides to isolate the term with ( x ):$$ y – 1 = 2x$$

Next, I

**divide**both sides by 2 to solve for ( x ):$$\frac{y – 1}{2} = x $$

Now, I interchange the **variables** to **invert** the function:

( x ) becomes the **input**, and ( y ) becomes the **output**. So, the **inverse function** is:

$$f^{-1}(x) = \frac{x – 1}{2} $$

The domain and range also swap roles in the **inverse function**. If the **domain** of the original function was all real numbers, then the range of the **inverse** is all real numbers, and vice versa.

Here’s a summary in a table for clarity:

Original Function | Inverse Function |
---|---|

( y = 2x + 1 ) | $ y = \frac{x – 1}{2}$ |

In terms of graphing, the graph of the **inverse function** is a reflection of the graph of the original function across the line ( y = x ).

Remember, all **linear functions** are **invertible** as long as their slope is not zero, which is the case with ( f(x) = 2x + 1 ). So finding the inverse is always possible for such functions, and it involves **algebraic manipulation** to **solve for y**.

## Applying Inverse Functions

When I encounter a function, such as ( f(x) = 2x + 1 ), finding the **inverse function** is like asking, “What operation would I need to apply to the output to get the original input?”

To **calculate** the **inverse**, I swap the ( x ) and ( y ), and then solve for ( y ). This is my starting point:

I begin by replacing ( f(x) ) with ( y ):

$$ y = 2x + 1$$I then interchange the variables:

$$x = 2y + 1$$Finally, I solve for ( y ) to find the

**inverse function**:

$$ x – 1 = 2y$$

$$ \frac{x – 1}{2} = y $$

So, the**inverse function**of ( f(x) ) is $ f^{-1}(x) = \frac{x – 1}{2}$.

To provide a practical **example**, if ( f(3) = 2(3) + 1 = 7 ), then using the **inverse function**, $ f^{-1}(7) = \frac{7 – 1}{2} = 3 $, which is our original input for ( f(x) ).

If I want to check my work quickly or handle more complicated functions, I might use a **function-inverse calculator**. These tools automate the process and give me a **result** without the manual algebra.

Function ((f(x))) | Inverse Function $(f^{-1}(x))$ |
---|---|

(2x + 1) | $\frac{x – 1}{2}$ |

By leveraging **inverse functions**, I can reverse operations and find original values, an essential concept in many mathematical applications.

## Conclusion

In this **exploration,** I’ve **determined** the **inverse** of the **function** **$$ f(x) = 2x + 1 $$**. To achieve this, the roles of the **input** and **output** were switched.

The process entailed setting **$$ y = f(x) $$** and **interchanging** x and y to solve for y. By **reversing** the **operations** of the **original function,** I arrived at the **inverse function $$ f^{-1}(x) = \frac{x – 1}{2} $$**.

It’s essential to verify that the **function** and its **inverse** truly undo each other’s operations. Applying** $$ f^{-1}(x) $$** to **$$ f(x) $$**, or vice versa, should return the original input value x.

In the context of our **function,** it means **substituting** **$$ f(x) $$** into the **inverse function** should get us back to **x**, confirming its correctness.

Lastly, finding the **inverse of a function** is not just a **mathematical exercise;** it has practical implications in various fields like **engineering, physics,** and **economics,** where one might need to revert to an **original** value after a set of **operations** has been applied.

The integrity of the original function and its **inverse** is therefore central to their utility in **real-world applications.**