Sure, let's find the inverse of the function [tex]\( f(x) = \frac{2}{3}x^2 - 6 \)[/tex].
1. Set the function equal to [tex]\(y\)[/tex]:
[tex]\[ y = \frac{2}{3} x^2 - 6 \][/tex]
2. Solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[ y + 6 = \frac{2}{3} x^2 \][/tex]
3. Multiply both sides by [tex]\(\frac{3}{2}\)[/tex] to isolate [tex]\(x^2\)[/tex]:
[tex]\[ \frac{3}{2}(y + 6) = x^2 \][/tex]
[tex]\[ x^2 = \frac{3}{2}(y + 6) \][/tex]
4. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{\frac{3}{2}(y + 6)} \][/tex]
5. Simplify the expression inside the square root:
[tex]\[ x = \pm \sqrt{\frac{3}{2}y + \frac{3}{2} \cdot 6} \][/tex]
[tex]\[ x = \pm \sqrt{\frac{3}{2}y + 9} \][/tex]
6. Rewrite as separate positive and negative solutions:
[tex]\[ x = 3 \sqrt{0.166666666666667y + 1} \][/tex]
[tex]\[ x = -3 \sqrt{0.166666666666667y + 1} \][/tex]
So, the inverse function [tex]\( f^{-1}(y) \)[/tex] is given by:
[tex]\[ f^{-1}(y) = 3 \sqrt{0.166666666666667y + 1} \][/tex]
or
[tex]\[ f^{-1}(y) = -3 \sqrt{0.166666666666667y + 1} \][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] can be written as:
[tex]\[ f^{-1}(x) = 3 \sqrt{0.166666666666667x + 1} \][/tex]
or
[tex]\[ f^{-1}(x) = -3 \sqrt{0.166666666666667x + 1} \][/tex]
