We are given the function [tex]\( f(x) = 2^x \)[/tex] and the transformed function [tex]\( g(x) = f(x + 2) \)[/tex].
To understand the key features of [tex]\( g(x) \)[/tex], let's examine how the transformation affects the original function [tex]\( f(x) \)[/tex].
### 1. Identifying the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. For [tex]\( g(x) = f(x + 2) \)[/tex], we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
g(0) = f(0 + 2) = f(2) = 2^2 = 4
\][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at [tex]\((0, 4)\)[/tex].
### 2. Identifying the horizontal asymptote of [tex]\( g(x) \)[/tex]:
The horizontal asymptote of an exponential function like [tex]\( f(x) = 2^x \)[/tex] is [tex]\( y = 0 \)[/tex]. The transformation [tex]\( f(x + 2) \)[/tex] is a horizontal shift to the left by 2 units, which does not affect the horizontal asymptote. Therefore, the horizontal asymptote for [tex]\( g(x) \)[/tex] remains:
[tex]\[
y = 0
\][/tex]
### 3. Determining the domain of [tex]\( g(x) \)[/tex]:
The domain of the original function [tex]\( f(x) = 2^x \)[/tex] is all real numbers, [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex]. A horizontal shift does not change the domain of the function. Therefore, the domain of [tex]\( g(x) = f(x + 2) \)[/tex] also is:
[tex]\[
\{ x \mid -\infty < x < \infty \}
\][/tex]
### Summary of key features:
- The [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\((0, 4)\)[/tex].
- The horizontal asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 0 \)[/tex].
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex].
Based on the provided options:
- [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex]: Incorrect.
- Horizontal asymptote of [tex]\( y = 2 \)[/tex]: Incorrect.
- Domain of [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex]: Correct.
- [tex]\( y \)[/tex]-intercept at [tex]\( (0, 4) \)[/tex]: Correct.
Thus, the correct statements describing key features of function [tex]\( g(x) \)[/tex] are that it has a [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex] and its domain is [tex]\(\{ x \mid -\infty < x < \infty \}\)[/tex].
