Answer :
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]
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]
