(a) Show that f is one-to-one. (b) Use Theorem 7 to find (f^-1)^'(a). (c) Calculate f^-1(x) and

step 1 Okay, so we're asked to show that our function f is one to one. So let's say that f evaluated at

(a) Show that f is one-to-one. (b) Use Theorem 7 to find...

(a) Show that $f$ is one-to-one. (b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$. (c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$. (d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes. $f(x)=9-x^2, \quad 0 \leqslant x \leqslant 3, \quad a=8$

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.
$f(x)=9-x^2, \quad 0 \leqslant x \leqslant 3, \quad a=8$

Key Concepts

Graphical Representation of Inverse Functions

Graphically, the inverse of a function can be represented by reflecting the original function's graph across the line y = x. This reflection property helps visualize the relationship between a function and its inverse. Sketching both graphs on the same set of axes not only demonstrates their symmetric nature but also reinforces the concept of reversibility between the input and output values.

Derivative of the Inverse Function

The derivative of an inverse function can be computed using the formula (f?¹)'(a) = 1 / f'(f?¹(a)), a result often stated as a theorem in calculus. This theorem provides a powerful tool for determining the rate of change of the inverse function at a point where f is differentiable and its derivative is nonzero, thereby connecting the behavior of a function to that of its inverse without requiring an explicit form of the inverse.

Domain and Range Considerations

When working with a function and its inverse, the domain of the original function becomes the range of its inverse, and vice versa. Correctly identifying these sets is critical in ensuring the well-defined nature of the inverse function. For example, when a function is given on a restricted domain to ensure one-to-one behavior, that domain restriction directly impacts the range of the function and subsequently the domain of the inverse.

One-to-One Functions

A one-to-one function, or injective function, is one in which every output is mapped from a unique input; that is, no two distinct inputs produce the same output. Demonstrating that a function is one-to-one often involves showing that if f(x1) = f(x2), then x1 must equal x2, or by showing that the function is strictly increasing or strictly decreasing on its domain. In many problems, especially when dealing with inverse functions, confirming one-to-one nature is crucial to ensure that an inverse exists.

Inverse Functions

An inverse function essentially reverses the roles of the inputs and outputs of the original function. If f is a one-to-one function, then its inverse, denoted as f?¹, satisfies the condition that f?¹(f(x)) = x and f(f?¹(y)) = y. The process of finding an inverse involves swapping variables and solving for the new dependent variable. Moreover, the domain and range of f and f?¹ are interchanged, which is an important aspect when establishing the inverse function completely.

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