Question

• The domain of the function is all real numbers.
• The range of the function is fy l y=0)-
• The y-intercept is 1.
• The horizontal asymptote is y=0, or the x-axis.
• The end behavior is:
as X→-∞,y→ co.
as X→ ∞,y→.

Answer 1

One simple function that meets all those conditions is
f(x) = e^{-x} (equivalently f(x) = 1/e^{x}).

Check:
- Domain: all real numbers (exponential defined for every x).
- Range: (0, ∞) (e^{-x} is always positive and can be arbitrarily large or arbitrarily close to 0).
- y‑intercept: f(0) = e^0 = 1.
- Horizontal asymptote: lim_{x→∞} e^{-x} = 0, so y = 0 is a horizontal asymptote.
- End behavior: lim_{x→-∞} e^{-x} = ∞ and lim_{x→∞} e^{-x} = 0.

If you want other functions with the same qualitative behavior but a different y‑intercept, you can use A·e^{-x} (A>0); to keep the y‑intercept 1 you must have A = 1.