- Swap the variables \(x\) and \(y\) in the function \(y = -\frac{1}{2}\sqrt{x + 3}\).
- Solve for \(y\) by multiplying by \(-2\), squaring both sides, and isolating \(y\).
- Determine the domain of the inverse function by analyzing the range of the original function, which results in \(x \leq 0\).
- Identify the constants in the inverse function: \(f^{-1}(x) = \boxed{4}x^2 - \boxed{3}\), for \(x \leq \boxed{0}\).
Explanation
- Analyzing the Original Function
To find the inverse function \(f^{-1}(x)\) of the given function \(f(x) = -\frac{1}{2}\sqrt{x + 3}\) with domain \(x \geq -3\), we follow a standard procedure of interchanging variables and solving for the dependent variable. First, observe that the range of the original function is \((-\infty, 0]\) because \(\sqrt{x + 3} \geq 0\), and multiplying by \(-\frac{1}{2}\) makes the function non-positive. This range will become the domain of the inverse function.
- Finding the Inverse Function
We start by setting \(y = -\frac{1}{2}\sqrt{x + 3}\) and then swap the roles of \(x\) and \(y\):
\[x = -\frac{1}{2}\sqrt{y + 3}\]
To solve for \(y\), first multiply both sides by \(-2\):
\[-2x = \sqrt{y + 3}\]
Since the square root must result in a non-negative value, we must have \(-2x \geq 0\), which implies \(x \leq 0\). This confirms the domain of our inverse function will be \(x \leq 0\). Next, square both sides to eliminate the square root:
\[(-2x)^2 = (\sqrt{y + 3})^2\]
\[4x^2 = y + 3\]
Solving for \(y\), we get:
\[y = 4x^2 - 3\]
Thus, the inverse function is \(f^{-1}(x) = 4x^2 - 3\) for \(x \leq 0\).
- Determining the Constants
Comparing our derived inverse function \(f^{-1}(x) = 4x^2 - 3\) for \(x \leq 0\) to the required form \(f^{-1}(x) = \square x^2 - \square, \text{ for } x \leq \square\), we identify the missing values as follows:
- The coefficient of \(x^2\) is \(4\).
- The constant subtracted is \(3\).
- The domain condition for \(x\) is \(x \leq 0\).
Therefore, the inverse function is \(f^{-1}(x) = 4x^2 - 3\) for \(x \leq 0\).
Examples
Inverse functions are essentially about reversing a process. Imagine a machine that takes an input number, adds 3, takes its square root, and then multiplies it by -1/2. An inverse machine would then take a result, reverse these steps one by one—reversing the -1/2 multiplication, then reversing the square root, and finally reversing the addition—to return to the original starting number. This is fundamental in fields like cryptography, where data is encoded (a function) and must be decoded (the inverse function) to retrieve the original message.