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Use the inverse of the function [tex]y = x^2 - 18x[/tex] to find the unknown values.

[tex]\[
\begin{array}{l}
b = \square \\
y = \pm \sqrt{bx + c} + d \\
c = \square \\
d = \square
\end{array}
\][/tex]


Sagot :

To find the unknown values in the inverse of the function \( y = x^2 - 18x \):

1. First, we look at the original function: \( y = x^2 - 18x \).

2. To find the inverse, we start by swapping \( x \) and \( y \):
[tex]\[ x = y^2 - 18y \][/tex]

3. We then arrange this as a quadratic equation in terms of \( y \):
[tex]\[ y^2 - 18y - x = 0 \][/tex]

4. To solve for \( y \), we use the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -18 \), and \( c = -x \). However, in this specific quadratic equation:
[tex]\[ y = \frac{18 \pm \sqrt{(18)^2 + 4x}}{2} \][/tex]

5. Simplifying the expression inside the square root and the equation itself:
[tex]\[ (18)^2 = 324 \implies y = \frac{18 \pm \sqrt{324 + 4x}}{2} \][/tex]
[tex]\[ y = \frac{18 \pm \sqrt{324 + 4x}}{2} = 9 \pm \sqrt{81 + x} \][/tex]

6. Thus, the inverse function is \( y = 9 \pm \sqrt{81 + x} \).

From this inverse function, we can see that the form is \( y = \pm \sqrt{bx + c} + d \). By comparison:

- \( b = 1 \)
- \( c = 81 \)
- \( d = 9 \)

So, the values are:
[tex]\[ \begin{array}{l} b = 1 \\ c = 81 \\ d = 9 \end{array} \][/tex]