- Analyze the domain and range of the function \(f(x) = \frac{\sqrt{x-2}}{6}\), identifying the domain as \(x \geq 2\) and range as \(y \geq 0\).
- Set \(y = f(x)\) and swap variables to obtain \(x = \frac{\sqrt{y-2}}{6}\).
- Isolate \(y\) by multiplying by 6, squaring both sides, and adding 2 to get \(y = 36x^2 + 2\).
- Conclude that the inverse is \(f^{-1}(x) = 36x^2 + 2\) for \(x \geq 0\), which is the first option.
- \(\boxed{f^{-1}(x) = 36x^2 + 2, \text{ for } x \geq 0}\)
Explanation
- Analyzing the Function and Its Domain/Range
To find the inverse of the function \(f(x) = \frac{\sqrt{x-2}}{6}\), we follow a standard algebraic procedure. First, we identify the domain of \(f(x)\). For the square root to be defined, the expression inside must be non-negative: \(x - 2 \geq 0\), which gives the domain \(x \geq 2\). The range of \(f(x)\) is \([0, \infty)\) because the square root function outputs non-negative values. The inverse function \(f^{-1}(x)\) will swap these: its domain will be \([0, \infty)\) (i.e., \(x \geq 0\)) and its range will be \([2, \infty)\).
- Finding the Inverse Function algebraically
To find the inverse function, we set \(y = f(x)\), which gives:
\[y = \frac{\sqrt{x-2}}{6}\]
To isolate the inverse relationship, we swap the variables \(x\) and \(y\):
\[x = \frac{\sqrt{y-2}}{6}\]
Now, we solve for \(y\):
Multiply both sides by 6 to get:
\[6x = \sqrt{y-2}\]
Square both sides of the equation to eliminate the square root:
\[(6x)^2 = y - 2\]
\[36x^2 = y - 2\]
Add 2 to both sides to solve for \(y\):
\[y = 36x^2 + 2\]
Thus, the inverse function is \(f^{-1}(x) = 36x^2 + 2\). Given our domain analysis, this is valid for \(x \geq 0\).
- Selecting the Correct Option
Comparing our derived result \(f^{-1}(x) = 36x^2 + 2\) for \(x \geq 0\) with the provided options, we see that it perfectly matches the first option. The function maps the non-negative real numbers to values at or above 2, which satisfies the requirements for the inverse of our original function. Therefore, the correct inverse is \(f^{-1}(x) = 36x^2 + 2\), for \(x \geq 0\).
Examples
Inverse functions are crucial in various fields, such as converting between different units of measurement. For instance, if you have a formula that calculates temperature in Fahrenheit based on Celsius, finding the inverse function allows you to quickly determine the temperature in Celsius given a Fahrenheit reading. Understanding how to find and verify these inverses ensures you can manipulate data accurately across different scales and systems.