Question

the function f(x) is shown below. is the inverse of f(x) also a function? use the drop - down menus to explain. f(x) = 2^x - 3, for all x click the arrows to choose an answer from each menu. the function f(x) choose... every range value to exactly one domain value. the domain of f(x) is choose... and the range of f(x) is choose.... therefore, the inverse of f(x) is choose... .

Explanation:

Step1: Determine if \( f(x) \) is one - to - one

A function \( y = f(x) \) is one - to - one if for any \( x_1
eq x_2\), \( f(x_1)
eq f(x_2)\). The function \( f(x)=2^{x}-3\) is an exponential function. The exponential function \( y = a^{x}\) (\(a>0,a
eq1\)) is one - to - one. For \( f(x)=2^{x}-3\), if \( x_1
eq x_2\), then \( 2^{x_1}
eq 2^{x_2}\), and \( 2^{x_1}-3
eq 2^{x_2}-3\). So \( f(x)\) maps every range value to exactly one domain value.

Step2: Find the domain and range of \( f(x)\)

The domain of an exponential function \( y = a^{x}+k\) (\(a > 0,a
eq1\)) is all real numbers. For \( f(x)=2^{x}-3\), the domain \(D_f=\mathbb{R}\) (all real numbers).
Since \( 2^{x}>0\) for all \(x\in\mathbb{R}\), then \(y = 2^{x}-3>- 3\). So the range \(R_f=(-3,\infty)\)

Step3: Determine the nature of the inverse

If a function \(y = f(x)\) is one - to - one (bijective, since it is also onto its range), then its inverse \(x = f^{-1}(y)\) is also a function.

Answer:

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