(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 let's start by showing that f is one to one. So assume there's two functions, f x1, whi

(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. . $f(x)=\sqrt{x-2}, \quad a=2$

(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.

. $f(x)=\sqrt{x-2}, \quad a=2$

Key Concepts

Graphical Reflections

The graph of a function and its inverse are symmetric with respect to the line y = x. This property is useful in visualizing and verifying the correctness of an inverse function, as plotting both can demonstrate how one graph 'mirrors' the other across this line.

Inverse Function Theorem

The Inverse Function Theorem provides a method for finding the derivative of the inverse function at a point. In general, if f is differentiable and its derivative is non-zero at a point, then the derivative of the inverse function can be computed as (f?¹)'(a) = 1 / f'(f?¹(a)). This theorem connects the behavior of a function and its inverse in terms of rates of change.

Domain and Range

When finding the inverse of a function, it is important to identify the domain of the original function and the range of the inverse, as these are interchanged. Properly determining these sets ensures that the inverse function is correctly defined and its applicable values are understood.

One-to-One Functions

A function is one-to-one (injective) if every output is produced by exactly one input. This property is crucial for the existence of an inverse function, as it ensures that the inverse function will also be well-defined, allowing a one-to-one correspondence between the domain and range of the function.

Inverse Functions

An inverse function reverses the operation of the original function, mapping outputs back to their corresponding inputs. If a function f is one-to-one, then its inverse, denoted as f?¹, will satisfy f(f?¹(x)) = x and f?¹(f(x)) = x within the appropriate domains.

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