Questions: Let f(x)=(x-6)^2 Find a domain on which f is one-to-one and non-decreasing. [6, ∞) Find the inverse of f restricted to this domain f^(-1)(x)=

Solution Steps

To find the inverse of the function \( f(x) = (x-6)^2 \) restricted to the domain \([6, \infty)\), we need to follow these steps:

1. Recognize that the function is one-to-one and non-decreasing on the given domain.
2. Solve the equation \( y = (x-6)^2 \) for \( x \) in terms of \( y \).
3. Ensure the solution for \( x \) falls within the given domain.

To find a domain on which \( f(x) = (x-6)^2 \) is one-to-one and non-decreasing, we need to analyze the behavior of the function.

The function \( f(x) = (x-6)^2 \) is a parabola that opens upwards with its vertex at \( x = 6 \). The function is non-decreasing for \( x \geq 6 \).

Thus, the domain on which \( f \) is one-to-one and non-decreasing is: \[ [6, \infty) \]

To find the inverse of \( f(x) = (x-6)^2 \) restricted to the domain \( [6, \infty) \), we follow these steps:

1. Replace \( f(x) \) with \( y \): \[ y = (x-6)^2 \]

2. Solve for \( x \) in terms of \( y \): \[ \sqrt{y} = x - 6 \] Since \( x \geq 6 \), we take the positive square root: \[ x = \sqrt{y} + 6 \]

3. Replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \sqrt{x} + 6 \]

Final Answer