To find the inverse function of [tex]\( g(x) = \frac{\sqrt{x} + 4}{1 - \sqrt{x}} \)[/tex], follow these steps:
### Step 1: Express the function in terms of [tex]\( y \)[/tex]
Let [tex]\( y = g(x) \)[/tex]. Thus,
[tex]\[ y = \frac{\sqrt{x} + 4}{1 - \sqrt{x}} \][/tex]
### Step 2: Solve for [tex]\( \sqrt{x} \)[/tex] in terms of [tex]\( y \)[/tex]
1. Multiply both sides by [tex]\( (1 - \sqrt{x}) \)[/tex] to get rid of the denominator:
[tex]\[ y(1 - \sqrt{x}) = \sqrt{x} + 4 \][/tex]
2. Distribute [tex]\( y \)[/tex] to both terms in the parentheses:
[tex]\[ y - y \sqrt{x} = \sqrt{x} + 4 \][/tex]
3. Rearrange the equation to collect all terms involving [tex]\( \sqrt{x} \)[/tex] on one side:
[tex]\[ y - 4 = y \sqrt{x} + \sqrt{x} \][/tex]
[tex]\[ y - 4 = \sqrt{x}(y + 1) \][/tex]
4. Solve for [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ \sqrt{x} = \frac{y - 4}{y + 1} \][/tex]
### Step 3: Square both sides to solve for [tex]\( x \)[/tex]
[tex]\[ x = \left(\frac{y - 4}{y + 1}\right)^2 \][/tex]
### Step 4: Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex]
To write the inverse function, replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ g^{-1}(x) = \left(\frac{x - 4}{x + 1}\right)^2 \][/tex]
Thus, the inverse function of [tex]\( g(x) = \frac{\sqrt{x} + 4}{1 - \sqrt{x}} \)[/tex] is:
[tex]\[ g^{-1}(x) = \left( \frac{x - 4}{x + 1} \right)^2 \][/tex]
