Question

Use composition of functions to show algebraically that your f′(x)
function from part a is the inverse of

Answer 1

To show that a function \( f'(x) \) is the inverse of a function \( f(x) \), we need to demonstrate that composing \( f \) and \( f' \) gives us the identity function. This means that:

1. \( f(f'(x)) = x \) for all \( x \) in the domain of \( f' \)
2. \( f'(f(x)) = x \) for all \( x \) in the domain of \( f \)

The identity function is defined as \( I(x) = x \), and it is crucial that both compositions yield \( x \).

### Step 1: Show \( f(f'(x)) = x \)

1. **Write \( f(y) \)**: If \( f(x) \) is a function we defined earlier, replace \( y \) with \( f'(x) \). The specific form of \( f'(x) \) will depend on \( f(x) \) and should be derived by differentiating \( f \).

2. **Calculate**: Substitute \( f'(x) \) into \( f(y) \) and manipulate the expression to confirm whether the result simplifies to \( x \).

### Step 2: Show \( f'(f(x)) = x \)

1. **Write \( f'(y) \)**: In this case, replace \( y \) with \( f(x) \).

2. **Calculate**: Substitute \( f(x) \) into \( f' \) and simplify. Check if the result simplifies to \( x \).

### Example:

Let's assume \( f(x) = x^2 \), then its derivative is \( f'(x) = 2x \).

- **Check \( f(f'((x))) \)**:

We need to substitute \( f'(x) \) into \( f(x) \):
\[
f(f'(x)) = f(2x) = (2x)^2 = 4x^2
\]
Here, we see that it does not simplify to \( x \) globally, so this is not a valid inverse at least in the direct sense.

- **Check \( f'(f(x)) \)**:

Now, let’s check the other direction and see if \( f'(f(x)) \) yields identity:
\[
f'(f(x)) = f'(x^2) = 2(x^2) = 2x^2
\]

In this case, neither check yields \( x \), so we see that \( f'(x) \) is not an inverse of \( f(x) \).

To correctly showcase an inverse relationship, the original function should maintain bijectivity (be one-to-one and onto). For instance, a suitable inverse might involve defining restrictions on the domain of \( f \), such as only allowing positive \( x \) for \( f(x) = x^2 \), or using the inverse function directly for inverses like \( f(x) = e^x \) where \( f^{-1}(x) = \ln(x) \).

### Conclusion:

To summarize, you would determine if indeed \( f'(x) \) can be established as the inverse of \( f(x) \) through careful composition and verification as shown in steps 1 and 2. Please provide specific functions if you are referring to particular \( f(x) \) and \( f'(x) \), so I can help apply this process directly!