Simplify the following inverse: $$ f\left(f^{-1}(x)\right)=[?] $$

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UpStudy AI · Accepted Answer

When you compose a function with its inverse, they effectively "undo" each other. Specifically:

$$
f\left(f^{-1}(x)\right) = x
$$

**Explanation:**

- **Function $$ f $$**: Takes an input $$ y $$ and maps it to $$ f(y) $$.
- **Inverse Function $$ f^{-1} $$**: Takes an input $$ x $$ (which is in the range of $$ f $$) and maps it back to $$ y $$, such that $$ f^{-1}(f(y)) = y $$.

When you compose $$ f $$ with $$ f^{-1} $$:

$$
f\left(f^{-1}(x)\right)
$$

Here's what happens step-by-step:

1. **Apply $$ f^{-1} $$ to $$ x $$**: This gives you the original input $$ y $$ such that $$ f(y) = x $$.
2. **Apply $$ f $$ to the result**: Applying $$ f $$ to $$ y $$ brings you back to $$ x $$.

So, the composition $$ f(f^{-1}(x)) $$ simplifies directly to $$ x $$, provided that $$ x $$ is within the domain where $$ f^{-1} $$ is defined.

**Final Simplified Form:**

$$
f\left(f^{-1}(x)\right) = x
$$
$$ f\left(f^{-1}(x)\right) = x $$

Simplify the following inverse:

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