- Complete the square for the function \(y = x^2 - 18x\) to get \(y = (x - 9)^2 - 81\).
- Isolate \(x\) to find the inverse relation, resulting in \(x = \pm \sqrt{y + 81} + 9\).
- Swap \(x\) and \(y\) to express the inverse function as \(y = \pm \sqrt{x + 81} + 9\).
- Match coefficients to determine \(b = 1\), \(c = 81\), and \(d = 9\).
- The final answer is \(\boxed{b=1, c=81, d=9}\).
Explanation
- Completing the Square and Finding the Inverse Relation
We start with the quadratic function \(y = x^2 - 18x\). To find the inverse, we express \(x\) in terms of \(y\). We complete the square for the quadratic expression:
\[y = x^2 - 18x\]
To complete the square, we add and subtract \((\frac{18}{2})^2 = 81\):
\[y = (x^2 - 18x + 81) - 81\]
\[y = (x - 9)^2 - 81\]
Now, isolate the term containing \(x\):
\[y + 81 = (x - 9)^2\]
Taking the square root of both sides gives:
\[\pm \sqrt{y + 81} = x - 9\]
Adding 9 to both sides, we find the expression for \(x\) in terms of \(y\):
\[x = \pm \sqrt{y + 81} + 9\]
The inverse relation is obtained by swapping the variables \(x\) and \(y\), which gives the inverse function \(y = \pm \sqrt{x + 81} + 9\).
- Identifying the Constants b, c, and d
Comparing the inverse function \(y = \pm \sqrt{x + 81} + 9\) with the given form \(y = \pm \sqrt{bx + c} + d\), we can identify the coefficients:
- The coefficient of \(x\) inside the square root is \(b = 1\).
- The constant term inside the square root is \(c = 81\).
- The constant term added outside the square root is \(d = 9\).
Thus, the values are \(b = 1\), \(c = 81\), and \(d = 9\).
Examples
Finding the inverse of a function is essential when you need to undo an operation. For instance, if you have a formula that calculates the area of a square from its side length, finding the inverse allows you to calculate the side length if you already know the area. In engineering or physics, if you have a mathematical model describing how a variable \(y\) changes with respect to \(x\), you can use the inverse relation to predict what input \(x\) is required to produce a desired output \(y\).