Question

What is the range of the function (e^x)/(e^x)+c? Assume that c is a constant greater than zero.

Answer 1

The range of the function (eˣ)/(eˣ+c) is the closed interval [0,1].

Step-by-step explanation:

The range of the function f(x) = (eˣ)/(eˣ+c) can be determined by analyzing the behavior of the function as x approaches positive and negative infinity. When x tends to positive infinity, the term eˣ grows much faster than c, so the denominator approaches infinity. Therefore, the function approaches 1 as x tends to infinity.

Similarly, when x tends to negative infinity, the term eˣ approaches 0 much faster than c, so the denominator becomes negligible. As a result, the function approaches 0 when x tends to negative infinity.

Since the function approaches 0 at negative infinity and approaches 1 at positive infinity, the range of the function is the closed interval [0,1].

Answer 2

Try this explanation:
1. if to re-write the given function as:

`y= (e^x)/(e^x+C); \ \ \textless \ =\ \textgreater \ \ y=1- (C)/(e^x+C);`
then it is possible to define its range:
2)

`\lim_(x \to+ \infty)[1- (C)/(e^x+C)]=1; \\ \lim_(x \to- \infty)[1- (C)/(e^x+C)]=0`

answer: (0;1)