Answer :
To determine the range of an exponential parent function with base 2, let's analyze the properties of exponential functions in general. An exponential function can be written in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential.
For an exponential function with base 2, we have:
[tex]\[ f(x) = 2^x \][/tex]
Here are the key properties of the exponential function [tex]\( f(x) = 2^x \)[/tex]:
1. Base Greater than 1: Since the base is 2, which is greater than 1, the function is increasing for all [tex]\( x \)[/tex] values.
2. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity: As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^x \)[/tex] approaches 0, but never actually reaches 0. Therefore, the function gets arbitrarily close to 0 but is always positive.
3. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity: As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 2^x \)[/tex] grows without bound.
4. No Negative Values: Since the base is positive and raised to any real number exponent, the function [tex]\( 2^x \)[/tex] never takes on negative values.
Given these observations, the possible values of [tex]\( y \)[/tex] (the outputs of [tex]\( f(x) \)[/tex]) are all positive real numbers. This can be written as:
[tex]\[ y > 0 \][/tex]
To restate, the outputs of the function [tex]\( 2^x \)[/tex] cover all positive real numbers, confirming that the range of this exponential parent function is the set of positive real numbers.
Thus, the correct answer is:
[tex]\[ \boxed{4 \text{ which corresponds to } D: \text{ Positive real numbers } (y>0)} \][/tex]
For an exponential function with base 2, we have:
[tex]\[ f(x) = 2^x \][/tex]
Here are the key properties of the exponential function [tex]\( f(x) = 2^x \)[/tex]:
1. Base Greater than 1: Since the base is 2, which is greater than 1, the function is increasing for all [tex]\( x \)[/tex] values.
2. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity: As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^x \)[/tex] approaches 0, but never actually reaches 0. Therefore, the function gets arbitrarily close to 0 but is always positive.
3. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity: As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 2^x \)[/tex] grows without bound.
4. No Negative Values: Since the base is positive and raised to any real number exponent, the function [tex]\( 2^x \)[/tex] never takes on negative values.
Given these observations, the possible values of [tex]\( y \)[/tex] (the outputs of [tex]\( f(x) \)[/tex]) are all positive real numbers. This can be written as:
[tex]\[ y > 0 \][/tex]
To restate, the outputs of the function [tex]\( 2^x \)[/tex] cover all positive real numbers, confirming that the range of this exponential parent function is the set of positive real numbers.
Thus, the correct answer is:
[tex]\[ \boxed{4 \text{ which corresponds to } D: \text{ Positive real numbers } (y>0)} \][/tex]
