Answer:
Option 1 is the correct answer.
Step-by-step explanation:
As per the curve given in the question, we can infer the following:
- Horizontal Asymptote (value of [tex]y[/tex] where [tex]x \rightarrow \infty[/tex]): as per graph horizontal asymptote is at [tex]y = 3[/tex].
- Vertical Asymptote (value of [tex]x[/tex] where [tex]y \rightarrow \infty[/tex]): as per graph vertical asymptote is at [tex]x = -2[/tex].
- [tex]y-[/tex] intercept (value of [tex]y[/tex] at which [tex]x = 0[/tex]): as per graph, [tex]y[/tex]-intercept is [tex]3.5[/tex].
- [tex]x-[/tex] intercept (value of [tex]x[/tex] at which [tex]y = 0[/tex]): as per graph, [tex]x[/tex]-intercept is [tex]-2.33[/tex].
- The values of [tex]x \text{ and } y[/tex] can be positive and negative, both.
Now, looking at the options given in the question figure:
Option 2 is exponential function i.e. [tex]2^{x}[/tex] which can never have negative value of [tex]y[/tex] so option 2 is not correct.
Option 3 has [tex]y[/tex]-intercept as 3. So option 3 is not correct.
Option 4 is logarithm function, so value of [tex]x\ in\ log x[/tex] can never be negative. So, option 4 is not correct.
When we see option 1, all the 5 conditions mentioned above (as interpreted from the question figure) are satisfied.
So, the equation of the figure is [tex]y = \frac{1}{x+2} + 3[/tex] i.e. option A is correct.
