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To find the inverse of a function means determining another function that `undoes` the effect of the original. For a function \( f(x) \), its inverse \( f^{-1}(x) \) switches the roles of inputs (x) and outputs (y). This implies that if \( f(a) = b \), then \( f^{-1}(b) = a \). Finding the inverse involves a series of steps:

• First, express the function \( y = f(x) \) in terms of y.
• Next, solve for \( x \) in terms of \( y \). This involves algebraic manipulation and sometimes taking square roots or other operations.
• Finally, swap the variables \( x \) and \( y \) to get \( f^{-1}(x) \).

For the function \( f(x) = x^2 - 2x + 1 \), which can be rewritten as \( f(x) = (x-1)^2 \), the inverse is found by setting \( y = (x-1)^2 \). Solving for \( x \), we derive \( x = \sqrt{y} + 1 \) which translates into the inverse function \( f^{-1}(x) = \sqrt{x} + 1 \), applicable where \( x \geq 0 \). This function \( \sqrt{x} + 1 \) reverses the original function's effect, returning to the initial input from its output.

The domain and range are crucial concepts when dealing with functions and their inverses. The **domain** of a function is the set of all possible input values (x-values), while the **range** is the set of possible output values (y-values). When finding the inverse of a function, the domain and range switch places.

For our given function \( f(x) = x^2 - 2x + 1 \), where \( x \geq 1 \), the domain is restricted to \( x \geq 1 \). The form \( f(x) = (x-1)^2 \) indicates that the range starts from 0 because \((x-1)^2\) cannot be negative. Thus, the range of \( f(x) \) is \( y \geq 0 \).

Inverting the function to \( f^{-1}(x) = \sqrt{x} + 1 \), the domain of the inverse becomes what the range of the original was: \( x \geq 0 \). Conversely, the range of the inverse is \( y \geq 1 \), as the smallest value \( \sqrt{x} + 1 \) can take is 1 (when \( x = 0 \)). Understanding these swapped roles assists in accurately identifying and working with inverse functions.

Quadratic functions are equations of the form \( ax^2 + bx + c \) and are characterized by their **parabolic** shapes. A key property is that they often include a vertex, the highest or lowest point on the graph, depending on the orientation. These functions can sometimes be rewritten in vertex form: \( a(x-h)^2 + k \), which reveals the vertex \( (h, k) \).

For the quadratic function \( f(x) = x^2 - 2x + 1 \), recognizing it in its transformed form \( (x-1)^2 \) indicates it's a perfect square trinomial. This makes finding the inverse easier as we directly express the transformation and find \( f^{-1} (x) = \sqrt{x} + 1 \).

Quadratic functions are **not automatically invertible** across their entire domain because they are not one-to-one (a one-to-one function has a unique x-value for each y-value). However, by restricting the domain, such as \( x \geq 1 \) in our example, these functions can have an inverse. These features highlight the versatility and distinct behavior of quadratic functions in algebra.