Let’s break down the function [tex]\( f(x) = \left(\frac{1}{2}\right)^x \)[/tex] to understand its domain, range, and behavior:
1. Domain of the function:
The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function [tex]\( f(x) = \left(\frac{1}{2}\right)^x \)[/tex], we can input any real number for [tex]\( x \)[/tex]. Thus,
- The domain of the function is all real numbers.
2. Range of the function:
The range of the function is the set of all possible output values (f(x) values). Since [tex]\( \left(\frac{1}{2}\right)^x \)[/tex] always produces positive values no matter the value of [tex]\( x \)[/tex] (since a positive fraction raised to any power is still positive), the values will approach zero but will never be zero. Therefore, the range of the function is all positive real numbers, not involving 'increases'.
3. Behavior of the function:
Since the base of our exponent, [tex]\( \frac{1}{2} \)[/tex], is a fraction less than 1, the function [tex]\( \left(\frac{1}{2}\right)^x \)[/tex] decreases as [tex]\( x \)[/tex] increases. This is because raising a fraction less than 1 to increasing powers yields progressively smaller results.
So, putting these observations together, the correct selections are:
- The domain of the function is all real numbers
- The function decreases as [tex]\( x \)[/tex] increases.
Thus, the completed statement should read:
"Consider the function [tex]\( f(x) = \left(\frac{1}{2}\right)^x \)[/tex].
Function [tex]\( f \)[/tex] has a domain of all real numbers and a range of positive real numbers. The function decreases as [tex]\( x \)[/tex] increases."
