To determine the range of the function [tex]\( y = 3^x \)[/tex], let's carefully analyze its behavior at different values of [tex]\( x \)[/tex]:
1. Behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \)[/tex] becomes very large negative, the exponentiation [tex]\( 3^x \)[/tex] will approach 0 but never actually reach 0. This is because the base 3 raised to an increasingly large negative power results in a very small positive fraction. Therefore, the output [tex]\( y \)[/tex] gets closer and closer to 0 from the positive side.
2. Behavior as [tex]\( x \to +\infty \)[/tex]:
As [tex]\( x \)[/tex] becomes very large positive, the value of [tex]\( 3^x \)[/tex] grows without bound. This means that the output [tex]\( y \)[/tex] can become arbitrarily large.
3. Behavior at intermediate values of [tex]\( x \)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( y = 3^0 = 1 \)[/tex].
For any other value of [tex]\( x > 0 \)[/tex], [tex]\( y = 3^x \)[/tex] is positive and greater than 1. As [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] increases exponentially.
For any value of [tex]\( x < 0 \)[/tex], [tex]\( y = 3^x \)[/tex] is still positive but less than 1, and as [tex]\( x \)[/tex] decreases further (i.e., becomes more negative), [tex]\( y \)[/tex] gets closer to 0 from the positive side.
From this analysis, we can conclude that the function [tex]\( y = 3^x \)[/tex] always produces positive real numbers regardless of whether [tex]\( x \)[/tex] is negative, zero, or positive.
Therefore, the range of the function [tex]\( y = 3^x \)[/tex] is all positive real numbers.
So, the range is:
positive real numbers (+R).
