Sure! Let's break down the solution step by step to understand why the range of the function [tex]\( f(x) = 4^x \)[/tex] is from zero to positive infinity.
1. Understanding the Function:
The given function is [tex]\( f(x) = 4^x \)[/tex]. This is an exponential function where the base is 4 and the exponent is [tex]\( x \)[/tex].
2. Properties of Exponential Functions:
Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) have the following properties:
- They are always positive for all real numbers [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] gets larger (approaches positive infinity), [tex]\( a^x \)[/tex] grows without bound (approaches positive infinity).
- As [tex]\( x \)[/tex] gets smaller (approaches negative infinity), [tex]\( a^x \)[/tex] approaches zero, but never actually reaches zero.
3. Behavior of [tex]\( 4^x \)[/tex]:
Applying these properties to [tex]\( f(x) = 4^x \)[/tex]:
- For any real number [tex]\( x \)[/tex], [tex]\( 4^x \)[/tex] is always positive because 4 is a positive number raised to any real power.
- When [tex]\( x \)[/tex] is a large positive number, [tex]\( 4^x \)[/tex] becomes very large. Thus, it tends towards positive infinity.
- When [tex]\( x \)[/tex] is a large negative number, [tex]\( 4^x \)[/tex] becomes very small, approaching zero but never reaching or becoming negative.
4. Range of the Function:
Considering the above behavior:
- The smallest value that [tex]\( 4^x \)[/tex] can get close to is zero, but it will not actually be zero.
- There is no upper limit to how large [tex]\( 4^x \)[/tex] can get, as it can grow indefinitely.
Therefore, the range of the function [tex]\( f(x) = 4^x \)[/tex] is from zero to positive infinity.
We can express the range in interval notation as [tex]\( (0, \infty) \)[/tex].
Hence, the range of the function is zero to positive infinity because [tex]\( 4^x \)[/tex] is always positive.
