To find the inverse function of [tex]\(g(x) = \frac{\sqrt{x} + 2}{7 - \sqrt{x}}\)[/tex], follow these steps:
1. Express the function in terms of [tex]\(y\)[/tex]:
[tex]\(y = \frac{\sqrt{x} + 2}{7 - \sqrt{x}}\)[/tex]
2. Interchange [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\(x = \frac{\sqrt{y} + 2}{7 - \sqrt{y}}\)[/tex]
3. Solve for [tex]\(y\)[/tex]:
- Start by manipulating the equation to clear the fraction. Multiply both sides by the denominator:
[tex]\(x(7 - \sqrt{y}) = \sqrt{y} + 2\)[/tex]
- Distribute [tex]\(x\)[/tex] on the left side:
[tex]\(7x - x\sqrt{y} = \sqrt{y} + 2\)[/tex]
- Isolate terms involving [tex]\(\sqrt{y}\)[/tex]:
[tex]\(7x - 2 = x\sqrt{y} + \sqrt{y}\)[/tex]
- Factor out [tex]\(\sqrt{y}\)[/tex] on the right side:
[tex]\(7x - 2 = \sqrt{y}(x + 1)\)[/tex]
- Divide both sides by [tex]\((x + 1)\)[/tex] to solve for [tex]\(\sqrt{y}\)[/tex]:
[tex]\(\sqrt{y} = \frac{7x - 2}{x + 1}\)[/tex]
- Square both sides to solve for [tex]\(y\)[/tex]:
[tex]\(y = \left(\frac{7x - 2}{x + 1}\right)^2\)[/tex]
Therefore, the inverse function of [tex]\(g(x)\)[/tex] is:
[tex]\[ g^{-1}(x) = \left(\frac{7x - 2}{x + 1}\right)^2 \][/tex]
This is the final expression for the inverse function.
