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Function verification is an essential step in confirming whether two functions are inverses.
When you find the inverse of a function, it is crucial to double-check the result using function verification.
This process ensures that when the inverse function is composed with the original function, you retrieve the input value back.

• First, let's understand what you need to verify: for the function \( f(x) = \frac{1}{3}x \), its inverse should satisfy two main conditions: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
• The idea is that applying a function and then its inverse should yield the identity, which means you get back the original input \( x \).

The inverse you found was \( f^{-1}(x) = 3x \).Start by verifying \( f(f^{-1}(x)) \):

• Substitute \( f^{-1}(x) \) into \( f \): the function becomes \( f(3x) = \frac{1}{3} \times 3x = x \).

This equation confirms the identity property.Next, verify \( f^{-1}(f(x)) \):

• Substitute \( f(x) \) into \( f^{-1} \): the function becomes \( f^{-1}\left(\frac{1}{3}x\right) = 3 \times \frac{1}{3}x = x \).

By checking both of these, you see that the inverse function correctly reverses the effect of the original function, confirming your solution is valid.

Algebraic manipulation is the backbone of finding inverse functions.
The goal is to change the structure of the equation to isolate the variable of interest.
For the function we are working with, \( f(x) = \frac{1}{3}x \), follow these straightforward algebraic steps to find its inverse:1. **Swap the roles of \( x \) and \( y \):** The original function can be written as \( y = \frac{1}{3}x \). To swap \( x \) and \( y \), write it as \( x = \frac{1}{3}y \). 2. **Solve for the new \( y \):** To make \( y \) a subject, multiply both sides by 3. This gives you the equation \( y = 3x \). Now, \( y \) represents the inverse function \( f^{-1}(x) \).Through these manipulations, you've effectively switched inputs and outputs, resulting in a new function that undoes the original function's operations.

Composition of functions is a method used to understand and verify the relationship between functions and their inverses.
When you compose two functions, you're essentially plugging one function into another.

• To verify inverse functions, you perform composition in two ways: \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \).
• For \( f(x) = \frac{1}{3}x \) and its inverse \( f^{-1}(x) = 3x \), both compositions should yield the identity function, meaning they return \( x \).

**Why do we do this?**- Composition is a double-check. It's crucial because even small errors in calculating an inverse can go unnoticed without this step.- It makes sure the function truly "undoes" the work of the original, bringing us back to our starting point.Performing these compositions confirms the functions are true inverses, solidifying our understanding and ability to manipulate and verify functions mathematically.