Question

what is the domain of the inverse of the function $f(x) = 13(0.75)^x + 1.73$?
a. $(13, \infty)$
b. $(1.73, \infty)$
c. $(1.73, 13)$
d. $(-\infty, \infty)$

Explanation:

Step1: Analyze the range of the original function

For the exponential function \(y = a\cdot b^{x}+k\) (\(a = 13\), \(b=0.75\), \(k = 1.73\)), since \(b = 0.75\in(0,1)\), \(y = 13(0.75)^{x}\gt0\). Then \(f(x)=13(0.75)^{x}+1.73\gt1.73\). The range of \(y = f(x)\) is \((1.73,\infty)\).

Step2: Use the property of inverse - function

The domain of the inverse function \(f^{-1}(x)\) is equal to the range of the original function \(y = f(x)\).

Answer:

B. \((1.73,\infty)\)