the function f(x) is shown below. is the inverse of f(x) also a function? use the drop - down menus to…

Explanation:

Step1: Check one - to - one property

The exponential function $y = 2^{x}$ is one - to - one. The function $f(x)=2^{x}-3$ is a vertical shift of $y = 2^{x}$ down by 3 units. A one - to - one function maps every range value to exactly one domain value. So, $f(x)$ maps every range value to exactly one domain value.

Step2: Find domain and range

For the function $f(x)=2^{x}-3$, the domain of an exponential function of the form $y = a^{x}+b$ (where $a>0,a\neq1$) is all real numbers, i.e., $(-\infty,\infty)$. Since $2^{x}>0$ for all $x\in R$, then $2^{x}-3>- 3$. So the range of $f(x)$ is $(-3,\infty)$.

Step3: Determine inverse

A function has an inverse that is also a function if and only if the original function is one - to - one. Since $f(x)$ is one - to - one, its inverse is also a function.

Answer:

The function $f(x)$ maps every range value to exactly one domain value. The domain of $f(x)$ is $(-\infty,\infty)$ and the range of $f(x)$ is $(-3,\infty)$. Therefore, the inverse of $f(x)$ is a function.