Which one of the following sets is a possible domain for the function with rule f(x)=(x+3)2−6𝑓(𝑥)=(𝑥+3)2−6 if the inverse function is to exist?

To determine which set could be a possible domain for the function $f(x) = (x + 3)^2 - 6$ such that the inverse function exists, we need to ensure that $f(x)$ is a one-to-one function over that domain. A function is one-to-one if it is either strictly increasing or strictly decreasing throughout its domain.

Step 1: Break Down the Problem

1. Identify the key characteristics of the function.
2. Determine the conditions for a one-to-one function.
3. Analyze different possible domains to find a suitable one.

Step 2: Relevant Concepts

• A function is one-to-one if it passes the horizontal line test (no horizontal line intersects the graph of the function more than once).
• To find the derivative of the function will help determine its monotonicity.

Step 3: Analysis and Detail

1. Finding the derivative:

   The derivative of the function can be calculated as follows: $f(x) = (x + 3)^2 - 6$ Using the power rule: $f'(x) = 2(x + 3)$

2. Determine where the function is increasing or decreasing:

   • Set the derivative to zero to find critical points: $2(x + 3) = 0 \implies x = -3$
   • Analyze intervals around $x = -3$:
     • For $x < -3$: $f'(x) < 0$ (decreasing)
     • For $x > -3$: $f'(x) > 0$ (increasing)

3. Choosing the domain:

   • To ensure the function is one-to-one, the possible domain should either be $(-\infty, -3]$ or $[-3, \infty)$.

Step 4: Verify and Summarize

• The function $f(x)$ is decreasing for $x < -3$ and increasing for $x > -3$.
• Therefore, valid domains that ensure the function is one-to-one are either $(-\infty, -3]$ or $[-3, \infty)$.

Final Answer

A possible domain for the function $f(x) = (x + 3)^2 - 6$ for the existence of its inverse function is $[-3, \infty)$.