If $$ h(x) = 2^{x-3} + 1 $$, determine the expression for its inverse.

Question

UpStudy AI · Accepted Answer

To find the inverse of the function $$ h(x) = 2^{x-3} + 1 $$, follow these steps:

1. **Replace $$ h(x) $$ with $$ y $$:**
   $$
   y = 2^{x-3} + 1
   $$

2. **Isolate the exponential term:**
   $$
   y - 1 = 2^{x-3}
   $$

3. **Take the logarithm with base 2 of both sides to solve for $$ x $$:**
   $$
   \log_2(y - 1) = x - 3
   $$

4. **Solve for $$ x $$:**
   $$
   x = \log_2(y - 1) + 3
   $$

5. **Express the inverse function by replacing $$ y $$ with $$ x $$:**
   $$
   h^{-1}(x) = \log_2(x - 1) + 3
   $$

**Final Answer:**
$$
h^{-1}(x) = \log_2(x - 1) + 3
$$
The inverse function is $$ h^{-1}(x) = \log_2(x - 1) + 3 $$.

If \( h(x) = 2^{x-3} + 1 \), determine the expression for its inverse.