91Ó°ÊÓ

A function is differentiable if it has a derivative at every point in its domain. This means that we can find a tangent to the curve of the function at any point, providing information about the rate of change at that point. Differentiability is a key property when analyzing if a function can have an inverse that is also differentiable.

For a function to be differentiable:

• The function must be continuous across its domain.
• There must be no sharp corners or cusps in the graph of the function.

Therefore, when checking if an inverse function is differentiable, we also need to look at the differentiability of the original function. In our case, since the function \(f(x) = x^5 + x - 1\) is a polynomial, it is differentiable for all real numbers.

A monotonic function is one that consistently increases or decreases over its domain. The significance of monotonicity in the context of inverse functions is that it ensures each output corresponds to exactly one input, making the function one-to-one.

Monotonic functions can be classified as:

• Strictly increasing: When the function always increases, and no two different inputs map to the same output.
• Strictly decreasing: When the function always decreases.

For our function \(f(x) = x^5 + x - 1\), analyzing the derivative \(f'(x) = 5x^4 + 1\), we observe that it is strictly positive, indicating the function is strictly increasing. Therefore, our function is monotonic, allowing us to confirm the existence of an inverse.

The derivative of an inverse function gives us the rate of change of the inverse relative to its original function. When a function is invertible and differentiable, its inverse is also differentiable, and we can find its derivative in terms of the derivative of the original function.

The formula for the derivative of the inverse function \(f^{-1}\) is:\[(g^{-1})'(d) = \frac{1}{g'(c)}\]where \(g(c) = d\) and \(c\) is such that \(d = g(c)\).
Applying this to our function, we calculate at points \(c = 0, 1, -1\):

• At \(c = 0\), where \(d = -1\), the derivative is \(\left(f^{-1}\right)'(-1) = \frac{1}{1} = 1\).
• At \(c = 1\), where \(d = 1\), the derivative is \(\left(f^{-1}\right)'(1) = \frac{1}{6}\).
• At \(c = -1\), where \(d = -3\), the derivative is \(\left(f^{-1}\right)'(-3) = \frac{1}{6}\).

A function is strictly increasing if for any two points \(x_1 < x_2\), we have \(f(x_1) < f(x_2)\). This unvarying upward trend ensures that the function does not take the same value twice, making it injective.

This property plays a crucial role in confirming the existence of an inverse function. For \(f(x) = x^5 + x - 1\), we found its derivative, \(f'(x) = 5x^4 + 1\), which is always positive for all real numbers. This indicates that the function is strictly increasing.

Due to this behavior, no horizontal line intersects the graph of \(f(x)\) more than once, guaranteeing that an inverse function exists and is unique. A strictly increasing function not only tells us about the one-to-one nature but also aids us in predicting the behavior of its inverse more confidently.