Let's find the inverse of the function [tex]\(y = x^2 - 18x\)[/tex] step by step to determine the unknown values [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex].
### Step-by-Step Solution to Find the Inverse:
1. Start with the function:
[tex]\[
y = x^2 - 18x
\][/tex]
2. Express this function in the standard quadratic form:
[tex]\[
y = ax^2 + bx + c
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -18\)[/tex], and [tex]\(c = 0\)[/tex].
3. Complete the square to find the inverse:
To complete the square, we rewrite the quadratic expression in a way that isolates a perfect square.
[tex]\[
y = x^2 - 18x
\][/tex]
Add and subtract the same value to complete the square: the value to add and subtract is [tex]\(\left(\frac{b}{2}\right)^2\)[/tex].
[tex]\[
y = x^2 - 18x + 81 - 81
\][/tex]
Where [tex]\(81\)[/tex] is [tex]\(\left(\frac{-18}{2}\right)^2\)[/tex].
[tex]\[
y = (x - 9)^2 - 81
\][/tex]
4. Isolate the perfect square term:
[tex]\[
y + 81 = (x - 9)^2
\][/tex]
5. Take the square root of both sides:
[tex]\[
\sqrt{y + 81} = x - 9 \quad \text{or} \quad -\sqrt{y + 81} = x - 9
\][/tex]
6. Isolate [tex]\(x\)[/tex]:
[tex]\[
x = \sqrt{y + 81} + 9 \quad \text{or} \quad x = -\sqrt{y + 81} + 9
\][/tex]
7. Express the inverse function:
The inverse can be written in terms of the original function's variable [tex]\(y\)[/tex]:
[tex]\[
f^{-1}(y) = \pm\sqrt{y + 81} + 9
\][/tex]
### Matching This with the Form [tex]\(y = \pm \sqrt{b x + c} + d\)[/tex]:
Now, we compare this form to our inverse function.
[tex]\[
f^{-1}(y) = \pm\sqrt{y + 81} + 9
\][/tex]
By comparing the equations, we identify the coefficients:
- [tex]\(b = 81\)[/tex]
- [tex]\(c = 0\)[/tex]
- [tex]\(d = 9\)[/tex]
Thus, the unknown values are:
[tex]\[
\begin{array}{l}
b = 81 \\
c = 0 \\
d = 9 \\
\end{array}
\][/tex]
