Question

Which answers describe the end behaviors of the function modeled by the graph?

f(x)=(12)x+1−2f(x)=(12)x+1−2
Select each correct answer.


As x decreases without bound, f(x) increases without bound.

As x increases without bound, f(x) approaches the line y=−2y=−2 .

As x decreases without bound, f(x) approaches the line y=−2y=−2 .

As x decreases without bound, f(x) decreases without bound.

Answer 1

Answer: Options 1 and 2 are correct.

Explanation:

Answer 2

Options 1 and 2 are correct.

Explanation:

The graph represents the function

`f(x)=((1)/(2))^(x+1)-2`

From the given graph it is noticed that the value of f(x) approaches to infinity as x approaches to negative infinity and the value of f(x) approaches to -2 as x approaches to positive infinity.

It can also represented by limits.

As x increases without bound

`lim_(x\rightarrow \infty)f(x)=lim_(x\rightarrow \infty)((1)/(2))^(x+1)-2`

Apply limit.

`lim_(x\rightarrow \infty)f(x)=((1)/(2))^(\infty+1)-2=0-2=-2`

As x decreases without bound

`lim_(x\rightarrow -\infty)f(x)=lim_(x\rightarrow -\infty)((1)/(2))^(x+1)-2`

Apply limit.

`lim_(x\rightarrow -\infty)f(x)=((1)/(2))^(-\infty+1)-2=-2=\infty-2=\infty`

Therefore as x decreases without bound, f(x) increases without bound and as x increases without bound, f(x) approaches the line y=−2. Options 1 and 2 are correct.