Question

what is the domain of the function $y = 2e^{x}$ graphed below?
all real numbers greater than 2
all positive real numbers
all negative real numbers
all real numbers

Explanation:

Step1: Recall the definition of domain

The domain of a function is the set of all possible input values (x - values) for which the function is defined.

Step2: Analyze the exponential function \(y = 2e^{x}\)

The exponential function \(y = e^{x}\) has the form \(y=a\cdot e^{kx}+c\) (here \(a = 2\), \(k = 1\), \(c=0\)). For any real - valued \(x\), the expression \(e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}=1 + x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\) is well - defined. There is no real number \(x\) for which \(e^{x}\) is undefined. When we multiply \(e^{x}\) by a non - zero constant \(2\) (in the function \(y = 2e^{x}\)), the domain does not change.

Step3: Check the graph

Looking at the graph, as \(x\) approaches \(-\infty\), \(y = 2e^{x}\to0\) (but \(x\) can take any value on the left - hand side of the \(x\) - axis), and as \(x\) approaches \(+\infty\), \(y = 2e^{x}\to+\infty\). There are no breaks or holes in the graph for any real \(x\) - values.

Answer:

all real numbers