To find the inverse of the given function [tex]\( f(x) = \log_2(x-3) + 7 \)[/tex], we need to follow a series of steps. Here’s the detailed, step-by-step solution:
### Step 1: Write the function
The given function is
[tex]\[ f(x) = \log_2(x-3) + 7 \][/tex]
### Step 2: Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]
[tex]\[ y = \log_2(x-3) + 7 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
To find the inverse function, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
1. Subtract 7 from both sides:
[tex]\[ y - 7 = \log_2(x-3) \][/tex]
2. Convert the logarithmic equation to its exponential form:
[tex]\[ x - 3 = 2^{y-7} \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 2^{y-7} + 3 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 2^{x-7} + 3 \][/tex]
### Step 4: Analyze the inverse function
[tex]\[ f^{-1}(x) = 2^{x-7} + 3 \][/tex]
#### Vertical Asymptote
The original function [tex]\( f(x) \)[/tex] has a vertical asymptote where the logarithmic argument is zero:
[tex]\[ x - 3 = 0 \implies x = 3 \][/tex]
For the inverse function, the logarithm component transforms the vertical asymptote into a horizontal asymptote for the inverse function.
#### Horizontal Asymptote
Since the exponential function [tex]\( 2^{x-7} \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex];
[tex]\[ \lim_{x \to -\infty} (2^{x-7} + 3) = 3 \][/tex]
Thus, the inverse function has a horizontal asymptote of:
[tex]\[ y = 3 \][/tex]
### Step 5: Domain and Range
For the original function [tex]\( f(x) \)[/tex]:
- The domain is [tex]\( (3, \infty) \)[/tex] since [tex]\( \log_2(x-3) \)[/tex] is defined for [tex]\( x > 3 \)[/tex].
- The range is [tex]\( (-\infty, \infty) \)[/tex] since logarithmic functions extend indefinitely in the vertical direction.
For the inverse function [tex]\( f^{-1}(x) \)[/tex]:
- The domain of [tex]\( f^{-1}(x) \)[/tex] is the range of [tex]\( f(x) \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( f^{-1}(x) \)[/tex] is the domain of [tex]\( f(x) \)[/tex], which is [tex]\( (3, \infty) \)[/tex].
### Final Answers
1. The inverse function is:
[tex]\[ f^{-1}(x) = 2^{x-7} + 3 \][/tex]
2. The inverse function has a horizontal asymptote of [tex]\( y = 3 \)[/tex].
3. The range of the original function is [tex]\( (-\infty, \infty) \)[/tex].
4. The inverse function is increasing on its domain of [tex]\( (-\infty, \infty) \)[/tex].
This step-by-step solution provides a clear path to finding the inverse of the function and analyzing its properties.
