Question

Could you please help me with this question

Answer 1

`\checkmark\ \ B. \ \left( (1)/(2) ,\infty \right)`

Explanation:

f is a continuous and an increasing function

Where its domain is (−∞ , ∞)

Then

The range of f is :

`(\lim_(x\rightarrow -\infty ) f\ ,\ \lim_(x\rightarrow \infty ) f\ )`

`\bullet \ \lim_(x\rightarrow -\infty ) f\ =\lim_(x\rightarrow -\infty ) (1)/(3) e^{\left( x-(2)/(3) \right) }+(1)/(2) = 0 +(1)/(2) =(1)/(2)`

`\bullet \ \lim_(x\rightarrow +\infty ) f\ =\lim_(x\rightarrow +\infty ) (1)/(3) e^{\left( x-(2)/(3) \right) }+(1)/(2) = (+\infty) +(1)/(2) =+\infty`

Then the range is :

`\left( (1)/(2) ,\infty \right)`