Answer :
Let's proceed to analyze the given function [tex]\( g(x) = 2(2)^x \)[/tex] and compare it to another function [tex]\( f(x) \)[/tex], based on certain characteristics like the y-intercept and end behavior.
### 1. Finding the y-intercept of [tex]\( g(x) \)[/tex]:
The y-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
For [tex]\( g(x) = 2(2)^x \)[/tex]:
[tex]\[ g(0) = 2(2)^0 = 2(1) = 2 \][/tex]
So, the y-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].
### 2. Determining the end behavior of [tex]\( g(x) \)[/tex]:
As [tex]\( x \)[/tex] approaches positive infinity:
[tex]\[ \text{As } x \to +\infty, 2^x \to +\infty \text{ (since } 2^x \text{ grows exponentially)} \][/tex]
[tex]\[ g(x) = 2(2^x) \to +\infty \text{ (because multiplying an exponentially growing function by a constant still results in exponential growth)} \][/tex]
So, as [tex]\( x \to +\infty \)[/tex], [tex]\( g(x) \to +\infty \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ \text{As } x \to -\infty, 2^x \to 0 \text{ (since the base is greater than 1)} \][/tex]
[tex]\[ g(x) = 2(2^x) \to 0 \text{ (because multiplying by 2, a constant, does not change the limit)} \][/tex]
So, as [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to 0 \)[/tex].
### 3. Conclusion and comparison:
By analyzing the characteristics:
- The y-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].
- The end behavior as [tex]\( x \to +\infty \)[/tex] is that [tex]\( g(x) \to +\infty \)[/tex].
- The end behavior as [tex]\( x \to -\infty \)[/tex] is that [tex]\( g(x) \to 0 \)[/tex].
Considering this and the correct statement from the given options, we can conclude:
[tex]\[ \text{The correct statement that compares the two functions is B. They have the same } y\text{-intercept and the same end behavior.} \][/tex]
Thus, the answer is Option B.
### 1. Finding the y-intercept of [tex]\( g(x) \)[/tex]:
The y-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
For [tex]\( g(x) = 2(2)^x \)[/tex]:
[tex]\[ g(0) = 2(2)^0 = 2(1) = 2 \][/tex]
So, the y-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].
### 2. Determining the end behavior of [tex]\( g(x) \)[/tex]:
As [tex]\( x \)[/tex] approaches positive infinity:
[tex]\[ \text{As } x \to +\infty, 2^x \to +\infty \text{ (since } 2^x \text{ grows exponentially)} \][/tex]
[tex]\[ g(x) = 2(2^x) \to +\infty \text{ (because multiplying an exponentially growing function by a constant still results in exponential growth)} \][/tex]
So, as [tex]\( x \to +\infty \)[/tex], [tex]\( g(x) \to +\infty \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ \text{As } x \to -\infty, 2^x \to 0 \text{ (since the base is greater than 1)} \][/tex]
[tex]\[ g(x) = 2(2^x) \to 0 \text{ (because multiplying by 2, a constant, does not change the limit)} \][/tex]
So, as [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to 0 \)[/tex].
### 3. Conclusion and comparison:
By analyzing the characteristics:
- The y-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 2 \)[/tex].
- The end behavior as [tex]\( x \to +\infty \)[/tex] is that [tex]\( g(x) \to +\infty \)[/tex].
- The end behavior as [tex]\( x \to -\infty \)[/tex] is that [tex]\( g(x) \to 0 \)[/tex].
Considering this and the correct statement from the given options, we can conclude:
[tex]\[ \text{The correct statement that compares the two functions is B. They have the same } y\text{-intercept and the same end behavior.} \][/tex]
Thus, the answer is Option B.
