What is the Inverse of $$ g(x)=\sqrt{5 x-2}+1 $$, for all $$ x \geq \frac{2}{5} ? $$

Question

UpStudy AI · Accepted Answer

To find the inverse of the function $$ g(x) = \sqrt{5x - 2} + 1 $$ for $$ x \geq \frac{2}{5} $$, follow these steps:

1. **Start with the original equation:**
   $$
   y = \sqrt{5x - 2} + 1
   $$

2. **Isolate the square root term:**
   $$
   y - 1 = \sqrt{5x - 2}
   $$

3. **Square both sides to eliminate the square root:**
   $$
   (y - 1)^2 = 5x - 2
   $$

4. **Solve for $$ x $$:**
   $$
   5x = (y - 1)^2 + 2
   $$
   $$
   x = \frac{(y - 1)^2 + 2}{5}
   $$

5. **Express the inverse function:**
   $$
   g^{-1}(y) = \frac{(y - 1)^2 + 2}{5}
   $$

6. **Replace $$ y $$ with $$ x $$ to write the inverse function in terms of $$ x $$:**
   $$
   g^{-1}(x) = \frac{(x - 1)^2 + 2}{5}
   $$

**Final Answer:**
$$
g^{-1}(x) = \frac{\, (x - 1)^{2} + 2\,}{\,5\,}
$$
The inverse function is $$ g^{-1}(x) = \frac{(x - 1)^2 + 2}{5} $$.

What is the Inverse of \( g(x)=\sqrt{5 x-2}+1 \), for all \( x \geq \frac{2}{5} ? \)