91Ó°ÊÓ

Understanding the domain and range of a function and its inverse is crucial.
The domain of a function represents all possible input values. For our function, this means looking at the expression under the square root sign and setting it to be non-negative:
\[ x + 3 \geq 0 \Rightarrow x \geq -3 \].
Therefore, the domain of \(f\) is:
\[ D_f = [-3, \infty) \].

Now, considering the range of the function, examine how the output values change.
As \(x\) increases from \(-3\) to \(\infty\), the output \(y = 2 \sqrt{x+3} - 5\) starts at \(-5\) and goes to \(\infty\). Thus, the range of \(f\) is:

\[ R_f = [-5, \infty) \].

Switching these for the inverse function, the domain of \(f^{-1}\) is the range of \(f\):
\[ D_{f^{-1}} = [-5, \infty) \]. Similarly, the range of \(f^{-1}\) is:
\[ R_{f^{-1}} = [-3, \infty) \].

By finding the domain and range properly, you can understand the complete behavior of the function.

Function transformation involves translating, stretching, or reflecting graphs.
For our given function:
\[ f(x) = 2 \sqrt{x + 3} - 5 \],
\the transformations include:

• Horizontal translation left by 3 units (due to \(+3\) under the square root)
• Vertical stretch by a factor of 2 (since the square root function is multiplied by 2)
• Vertical translation down by 5 units (because of \(-5\) outside the square root)

Each transformation changes the graph's appearance in predictable ways.

Let's break them down:

• Start with the basic square root function \( \sqrt{x} \).
• Shift it left by 3 to get \( \sqrt{x+3} \).
• Stretch it vertically to form \( 2\sqrt{x+3} \).
• Finally, shift it down 5 units to find \( 2\sqrt{x+3} - 5 \).

Understanding these fundamental transformations helps you visualize and sketch the function accurately.

Square root functions naturally have specific characteristics and behaviors.
The most basic square root function is \( f(x) = \sqrt{x} \). This function has:

• A domain of \[ [0, \infty) \]
• A range of \[ [0, \infty) \]

Additions and multiplications to this function, like in our problem, transform its graph.
Let's review the given function
\[ f(x) = 2 \sqrt{x + 3} - 5 \]:

• The domain changes based on the expression inside the square root.
  For example, \( x + 3 \) must be non-negative, setting limitations on \(x\).
• The range is influenced by any value outside the square root, such as the \(-5\).
• The multiplier in front (like the 2) changes the steepness of the curve, but the basic shape remains the same.

The insights gained from transforming the basic square root function help in analyzing more complex functions easily.