What is the range of $$ f(x)=6-e^{-x} $$ ? $$ \square $$ (Type your answer in interval notation

Question

UpStudy AI · Accepted Answer

1. The function is given by 
   $$
   f(x)=6-e^{-x}.
   $$
2. Since the exponential function satisfies $$e^{-x} > 0$$ for all $$x \in \mathbb{R}$$, we have:
   $$
   e^{-x} \in (0, \infty).
   $$
3. As a result, 
   $$
   f(x)=6-e^{-x}
   $$
   takes values by subtracting a positive number from 6. The smallest value of $$f(x)$$ occurs when $$e^{-x}$$ is as large as possible.
4. As $$x \to -\infty$$, $$e^{-x} \to \infty$$. Therefore,
   $$
   f(x)=6-e^{-x} \to -\infty.
   $$
5. The largest value of $$f(x)$$ occurs when $$e^{-x}$$ is as small as possible, which is achieved as $$x \to \infty$$. In this case, $$e^{-x} \to 0$$ and 
   $$
   f(x) \to 6.
   $$
6. However, $$f(x)$$ never equals 6 because $$e^{-x}$$ is always positive. Thus, 6 is not included in the range.

The range of the function is 
$$
(-\infty, 6).
$$
The range of $$ f(x) = 6 - e^{-x} $$ is all real numbers less than 6, written as $$ (-\infty, 6) $$.

What is the range of ?

(Type your answer in interval notation

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Pre Calculus Questions

What is the range of ? (Type your answer in interval notation