To find the inverse of the function [tex]\( f(y) = \frac{3y + 2}{y^2 - 4} \)[/tex], we need to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. Let's go through the step-by-step process.
1. Set up the equation: We start by setting [tex]\( f(y) \)[/tex] equal to [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3y + 2}{y^2 - 4}
\][/tex]
2. Clear the fraction: To eliminate the fraction, multiply both sides by [tex]\( y^2 - 4 \)[/tex]:
[tex]\[
x(y^2 - 4) = 3y + 2
\][/tex]
3. Simplify the equation: Distribute [tex]\( x \)[/tex] on the left-hand side.
[tex]\[
xy^2 - 4x = 3y + 2
\][/tex]
4. Rearrange the terms: Move all terms to one side of the equation to set it to zero.
[tex]\[
xy^2 - 3y - 4x - 2 = 0
\][/tex]
5. Solve the quadratic equation for [tex]\( y \)[/tex]: This is a quadratic equation in [tex]\( y \)[/tex], and can be written in the standard form [tex]\( ay^2 + by + c = 0 \)[/tex], with:
[tex]\[
a = x, \quad b = -3, \quad c = -4x - 2
\][/tex]
6. Apply the quadratic formula: The quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] helps us solve for [tex]\( y \)[/tex]. Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(x)(-4x - 2)}}{2x}
\][/tex]
Simplify inside the square root:
[tex]\[
y = \frac{3 \pm \sqrt{9 + 16x^2 + 8x}}{2x}
\][/tex]
7. Final solutions: We obtain two possible solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3 + \sqrt{16x^2 + 8x + 9}}{2x} \quad \text{and} \quad y = \frac{3 - \sqrt{16x^2 + 8x + 9}}{2x}
\][/tex]
Thus, the inverse function for [tex]\( f(y) \)[/tex] is given by the two possible expressions:
[tex]\[
f^{-1}(x) = \frac{3 + \sqrt{16x^2 + 8x + 9}}{2x} \quad \text{or} \quad f^{-1}(x) = \frac{3 - \sqrt{16x^2 + 8x + 9}}{2x}
\][/tex]
