91Ó°ÊÓ

In calculus, a derivative measures how a function changes as its input changes. For the function given in the exercise, \( f(x) = \frac{1}{3}x + 2 \), we calculate its derivative to determine if the function is one-to-one.

We compute the derivative, \( f'(x) \), as follows:
\( f'(x) = \frac{d}{dx} \left( \frac{1}{3}x + 2 \right) = \frac{1}{3} \).

This derivative is a constant, which means it does not change as \( x \) changes. A constant and positive derivative, such as \( \frac{1}{3} \), signifies that our function has a consistent rate of increase.

In simpler terms, for every increase in \( x \), the function \( f(x) \) increases by the same amount. This constant rate of increase helps us in determining the one-to-one nature of the function.

An inverse function reverses the operation done by the original function. For the function \( f(x) = \frac{1}{3}x + 2 \), finding its inverse involves a few steps.

First, we set \( y = f(x) \):
* \( y = \frac{1}{3}x + 2 \)

Next, we solve for \( x \) in terms of \( y \):
* Subtract 2 from both sides: \( y - 2 = \frac{1}{3}x \)
* Multiply by 3 to isolate \( x \): \( x = 3(y - 2) \)

Finally, we express the inverse function by switching roles: replace \( y \) with \( x \). Thus, the inverse function is:
* \( f^{-1}(x) = 3(x - 2) \)

This inverse function means if you input a value into \( f^{-1}(x) \), you get back the original \( x \) that was put into \( f(x) \) to get that value.

A strictly increasing function is a function that always rises as the input value increases. For our function, \( f(x) = \frac{1}{3}x + 2 \), we determine if it's strictly increasing by looking at its derivative.

As calculated earlier, the derivative \( f'(x) = \frac{1}{3} \) is always positive. Since the derivative is positive for all values of \( x \), the function is consistently rising and thus strictly increasing.

This property is important because it's what makes \( f(x) \) one-to-one. In a one-to-one function, every output value corresponds to exactly one input value. With our strictly increasing function, no two different \( x \) values can produce the same \( f(x) \) value, ensuring the one-to-one nature.