Let's analyze the parent function [tex]\( f(x) = e^x \)[/tex] and the transformed function [tex]\( g(x) = f(x+3) = e^{x+3} \)[/tex] in detail to determine which feature differs between them.
1. Y-Intercept:
- The y-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) \)[/tex], we have:
[tex]\[
f(0) = e^0 = 1
\][/tex]
- For [tex]\( g(x) \)[/tex], we have:
[tex]\[
g(0) = e^{0+3} = e^3 \approx 20.085536923187668
\][/tex]
- Therefore, the y-intercepts are different: [tex]\( f(x) \)[/tex] has a y-intercept of [tex]\( 1 \)[/tex] and [tex]\( g(x) \)[/tex] has a y-intercept of approximately [tex]\( 20.085536923187668 \)[/tex].
2. Range:
- The range of the parent function [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex] since [tex]\( e^x \)[/tex] is always positive for all real [tex]\( x \)[/tex].
- Transforming [tex]\( f(x) \)[/tex] by shifting it horizontally does not change the range.
- Thus, the range of [tex]\( g(x) = e^{x+3} \)[/tex] is also [tex]\( (0, \infty) \)[/tex].
- So, the range remains the same for both functions.
3. Domain:
- The domain of [tex]\( f(x) = e^x \)[/tex] is all real numbers [tex]\( (-\infty, \infty) \)[/tex] because [tex]\( e^x \)[/tex] is defined for all real [tex]\( x \)[/tex].
- Similarly, [tex]\( g(x) = e^{x+3} \)[/tex] is also defined for all real [tex]\( x \)[/tex].
- Thus, the domain remains the same for both functions.
4. Horizontal Asymptote:
- The horizontal asymptote of [tex]\( f(x) = e^x \)[/tex] is [tex]\( y = 0 \)[/tex], since as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\( e^x \)[/tex] approaches [tex]\( 0 \)[/tex].
- Shifting [tex]\( f(x) \)[/tex] horizontally to obtain [tex]\( g(x) \)[/tex] does not affect the horizontal asymptote.
- Therefore, the horizontal asymptote of [tex]\( g(x) = e^{x+3} \)[/tex] is also [tex]\( y = 0 \)[/tex].
Given this analysis, the feature that differs between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is the y-intercept. Thus, the correct answer is:
[tex]$y$[/tex]-Intercept
