91Ó°ÊÓ

A function is one-to-one (or injective) if each element in the range corresponds to exactly one element in the domain. In other words, no two different inputs produce the same output. This is a crucial property because only one-to-one functions have inverses that are also functions.
To check if a function is one-to-one, you can use the horizontal line test. If any horizontal line intersects the graph of the function at most once, then the function is one-to-one.
In our exercise, the function given is \( g(x) = \sqrt{x - 3} \) with the restriction that \( x \geq 3 \). This ensures that the function passes the horizontal line test. Thus, we can conclude it is a one-to-one function and it has an inverse.

To find the inverse of a function, we need to solve the equation for the input variable. Begin by replacing the function notation with a new variable. For \( g(x) = \sqrt{x - 3} \), we replace \( g(x) \) with \( y \) to get the equation \( y = \sqrt{x - 3} \).
The goal is to express \( x \) in terms of \( y \). To do this, undo the operations in reverse order. Since the function involves a square root, we start by squaring both sides: \( y^2 = x - 3 \).
Next, isolate \( x \) by adding 3 to both sides of the equation: \( x = y^2 + 3 \). Now, \( x \) is expressed in terms of \( y \), setting us up to write the inverse function.

Algebraic manipulation involves using mathematical operations to transform an equation into a desired form.
Once we have isolated \( x \), we need to express the inverse function. Replace \( y \) with \( x \) to write the inverse function \( g^{-1}(x) \): \( g^{-1}(x) = x^2 + 3 \).
We also need to consider the domain of the original function. Since the original function operates for \( x \geq 3 \), the range of the function \( g(x) \) is \([0, \infty)\). Therefore, the domain of the inverse function must also be \( x \geq 0 \).
To summarize, the inverse function is \( g^{-1}(x) = x^2 + 3 \) for \( x \geq 0 \). This ensures the inverse function adheres to the same restrictions as the original function, maintaining the one-to-one relationship.