Inverse Functions - Understand in 10 Minutes - Free Printable

Great! Let's solve this step-by-step and understand inverse functions using the given information:

We are told:

> $ f^{-1}(x) = 2x - 1 $

Our goal is to understand inverse functions and possibly find the original function $ f(x) $, since we're given its inverse.

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🔍 What is an Inverse Function?

An inverse function, denoted $ f^{-1}(x) $, "undoes" what the original function $ f(x) $ does.

- If $ f(a) = b $, then $ f^{-1}(b) = a $
- The composition of a function and its inverse gives the identity:
$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
$$

So if we know $ f^{-1}(x) = 2x - 1 $, we can find $ f(x) $ by solving for the original function.

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🧠 Step-by-Step: Find $ f(x) $ from $ f^{-1}(x) $

Let’s suppose:
$$
f^{-1}(x) = 2x - 1
$$

To find $ f(x) $, we reverse the process.

Let’s say:
$$
y = f^{-1}(x) = 2x - 1
$$

But in terms of inverses, if $ y = f^{-1}(x) $, then $ x = f(y) $. So we want to express $ x $ in terms of $ y $, and then swap variables.

#### Step 1: Start with the inverse
$$
y = 2x - 1
$$

#### Step 2: Solve for $ x $ in terms of $ y $
$$
y = 2x - 1 \\
\Rightarrow y + 1 = 2x \\
\Rightarrow x = \frac{y + 1}{2}
$$

Now, since $ x = f(y) $, this means:
$$
f(y) = \frac{y + 1}{2}
$$

Now replace $ y $ with $ x $ (standard notation):
$$
f(x) = \frac{x + 1}{2}
$$

✔ So, the original function is:
$$
f(x) = \frac{x + 1}{2}
$$

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✔ Check: Verify that $ f $ and $ f^{-1} $ are inverses

Let’s check $ f(f^{-1}(x)) = x $

Compute:
$$
f(f^{-1}(x)) = f(2x - 1) = \frac{(2x - 1) + 1}{2} = \frac{2x}{2} = x
$$

Good!

Now check $ f^{-1}(f(x)) = x $:
$$
f^{-1}(f(x)) = f^{-1}\left(\frac{x + 1}{2}\right) = 2\left(\frac{x + 1}{2}\right) - 1 = (x + 1) - 1 = x
$$

Perfect! They are inverses.

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📌 Summary

Given:
$$
f^{-1}(x) = 2x - 1
$$

We found:
$$
f(x) = \frac{x + 1}{2}
$$

And verified they undo each other.

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💡 Key Takeaway: How to Find Inverse Functions

To find $ f^{-1}(x) $ from $ f(x) $:

1. Replace $ f(x) $ with $ y $
2. Swap $ x $ and $ y $
3. Solve for $ y $
4. Replace $ y $ with $ f^{-1}(x) $

In reverse, if you’re given $ f^{-1}(x) $, you can find $ f(x) $ by reversing the steps.

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🎯 Final Answer:

The original function is:
$$
\boxed{f(x) = \frac{x + 1}{2}}
$$

And this is the inverse of $ f^{-1}(x) = 2x - 1 $. ✔

Let me know if you'd like a graph or more examples!