To determine the range of the function [tex]\( y = 4e^x \)[/tex], we need to understand how the function behaves as [tex]\( x \)[/tex] varies over all real numbers.
1. Understanding the exponential function [tex]\( e^x \)[/tex]:
- The exponential function [tex]\( e^x \)[/tex] is always positive for all real [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( e^x \)[/tex] approaches 0.
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]), [tex]\( e^x \)[/tex] approaches infinity.
2. Applying the behavior of [tex]\( e^x \)[/tex] to [tex]\( y = 4e^x \)[/tex]:
- Since [tex]\( y = 4e^x \)[/tex], the value of [tex]\( y \)[/tex] will be 4 times the value of [tex]\( e^x \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( e^x \)[/tex] approaches 0, thus [tex]\( y \)[/tex] approaches [tex]\( 4 \times 0 = 0 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( e^x \)[/tex] approaches infinity, thus [tex]\( y \)[/tex] approaches [tex]\( 4 \times \infty = \infty \)[/tex].
3. Range of [tex]\( y = 4e^x \)[/tex]:
- Since [tex]\( y \)[/tex] approaches 0 but never actually reaches 0, the smallest value for [tex]\( y \)[/tex] is greater than 0.
- Since [tex]\( y \)[/tex] can grow without bound as [tex]\( x \)[/tex] increases, the values of [tex]\( y \)[/tex] are all positive numbers greater than 0.
Therefore, the range of the function [tex]\( y = 4e^x \)[/tex] is all real numbers greater than 0.
[tex]\[
\text{The range of the function } y = 4e^x \text{ is all real numbers greater than 0.}
\][/tex]
