- To find the inverse, replace \(y\) with \(x\) and \(x\) with \(y\) to get \(x = 16y^2 + 1\).
- Isolate \(y^2\) by subtracting 1 and dividing by 16, resulting in \(y^2 = \frac{x - 1}{16}\).
- Take the square root to solve for \(y\), yielding \(y = \pm \frac{\sqrt{x - 1}}{4}\).
- The correct option is \(\boxed{a}\).
Explanation
- Understanding the Inverse Relation
To find the inverse of the function \(y = 16x^2 + 1\), we follow a standard algebraic procedure. An inverse function reverses the mapping of the original function. We start by writing the given equation:
\[y = 16x^2 + 1\]
The original function has a domain of all real numbers \(x\) and a range of \(y \geq 1\). When we determine the inverse, the domain of the new relation will be \(x \geq 1\), and its range will be all real numbers.
- Solving for y
To find the inverse, we swap the roles of \(x\) and \(y\). Replacing \(y\) with \(x\) and \(x\) with \(y\) gives:
\[x = 16y^2 + 1\]
Now, we solve this equation for \(y\). First, subtract 1 from both sides of the equation:
\[x - 1 = 16y^2\]
Next, divide both sides by 16:
\[y^2 = \frac{x - 1}{16}\]
Finally, take the square root of both sides to solve for \(y\):
\[y = \pm \sqrt{\frac{x - 1}{16}}\]
- Simplifying the Expression and Conclusion
We can simplify the expression for \(y\) by using the properties of square roots:
\[y = \pm \frac{\sqrt{x - 1}}{\sqrt{16}}\]
Since the square root of 16 is 4, the expression becomes:
\[y = \pm \frac{\sqrt{x - 1}}{4}\]
Comparing this result to the given options, we find it matches option (a). Thus, the inverse of the function is \(y = \frac{\pm \sqrt{x - 1}}{4}\).
Examples
Finding inverse functions is essential in various fields, such as computer graphics and cryptography. For instance, if a computer program uses a specific function to encode data, the decoding process often requires the inverse function to retrieve the original information. Similarly, in economics, if a function relates the price of an item to the quantity demanded, the inverse function allows companies to determine the price required to achieve a specific level of demand. Understanding how to algebraically manipulate equations to find inverses is a foundational skill for these and many other practical applications.