To solve this problem, let's first understand the key features of the function [tex]\( f(x) = 2^x \)[/tex] and how the transformation affects it.
The function [tex]\( f(x) = 2^x \)[/tex] is an exponential function. A key feature of exponential functions is the y-intercept, which is the point where the graph of the function crosses the y-axis. For [tex]\( f(x) = 2^x \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2^0 = 1 \][/tex]
So, the y-intercept of [tex]\( f(x) = 2^x \)[/tex] is at the point [tex]\((0, 1)\)[/tex].
Now, let's consider the function [tex]\( g(x) = 2 f(x) \)[/tex]. This means that [tex]\( g(x) \)[/tex] is a vertically stretched version of [tex]\( f(x) \)[/tex] by a factor of 2. Mathematically, this represents:
[tex]\[ g(x) = 2 \cdot 2^x = 2^{x+1} \][/tex]
To find the y-intercept of [tex]\( g(x) \)[/tex], we need to determine the value of [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 2 \cdot f(0) = 2 \cdot 1 = 2 \][/tex]
So, the y-intercept of [tex]\( g(x) = 2 f(x) \)[/tex] is at the point [tex]\((0, 2)\)[/tex].
Thus, among the given options, the correct answer is:
B. [tex]\( y \)[/tex]-intercept at [tex]\((0, 2)\)[/tex]
