Answer:
[tex]\checkmark\ \ B. \ \left( \frac{1}{2} ,\infty \right)[/tex]
Step-by-step explanation:
f is a continuous and an increasing function
Where its domain is (−∞ , ∞)
Then
The range of f is :
[tex](\lim_{x\rightarrow -\infty } f\ ,\ \lim_{x\rightarrow \infty } f\ )[/tex]
[tex]\bullet \ \lim_{x\rightarrow -\infty } f\ =\lim_{x\rightarrow -\infty } \frac{1}{3} e^{\left( x-\frac{2}{3} \right) }+\frac{1}{2} = 0 +\frac{1}{2} =\frac{1}{2}[/tex]
[tex]\bullet \ \lim_{x\rightarrow +\infty } f\ =\lim_{x\rightarrow +\infty } \frac{1}{3} e^{\left( x-\frac{2}{3} \right) }+\frac{1}{2} = (+\infty) +\frac{1}{2} =+\infty[/tex]
Then the range is :
[tex]\left( \frac{1}{2} ,\infty \right)[/tex]
