To answer the question, let's start by finding the inverse of the function given by [tex]\( y = x^2 - 18x \)[/tex].
### Step 1: Find the Inverse of [tex]\( y = x^2 - 18x \)[/tex]
First, we need to rewrite the equation in terms of [tex]\( x \)[/tex]:
[tex]\[ y = x^2 - 18x \][/tex]
To find the inverse, solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x^2 - 18x - y = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. We can solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = -y \)[/tex]. Substituting in these values:
[tex]\[ x = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{18 \pm \sqrt{324 + 4y}}{2} \][/tex]
[tex]\[ x = \frac{18 \pm \sqrt{4(y + 81)}}{2} \][/tex]
[tex]\[ x = 9 \pm \sqrt{y + 81} \][/tex]
Therefore, there are two possible functions for the inverse:
[tex]\[ x = 9 + \sqrt{y + 81} \quad \text{or} \quad x = 9 - \sqrt{y + 81} \][/tex]
Each of these corresponds to one of the branches of the inverse function.
### Step 2: Determine values [tex]\( b, c, d \)[/tex]
We are given:
[tex]\[ b = \pm \sqrt{b x + c} + d \][/tex]
Let’s compare this form with one of the inverse functions found. We choose:
[tex]\[ x = 9 + \sqrt{y + 81} \][/tex]
It's clear that [tex]\( \sqrt{y + 81} \)[/tex] can be rewritten in the form [tex]\( \sqrt{b x + c} \)[/tex] by comparing terms.
We know that:
[tex]\[ \sqrt{y + 81} \][/tex]
should match with:
[tex]\[ \sqrt{b x + c} \][/tex]
So,
[tex]\[ y + 81 = b x + c \][/tex]
Using the inverse form [tex]\( x = 9 + \sqrt{y + 81} \)[/tex], we deduce:
[tex]\[ y + 81 = x - 9 \][/tex]
Then in the standardized form:
[tex]\[ y + 81 = 1 (x - 9) \][/tex]
So, this implies:
[tex]\[ b = 1 \][/tex]
[tex]\[ c = -9 \][/tex]
Finally, to match the inverse correctly, the outer term, [tex]\( \pm \sqrt{y + 81} + 9 \)[/tex], implies:
[tex]\[ d = 9 \][/tex]
Therefore, the values are:
[tex]\[
\begin{array}{l}
b = 1 \\
c = -9 \\
d = 9 \\
\end{array}
\][/tex]
