The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)).

10) If \( f(x)=\frac{1}{x-a}+b \), then \( f^{-1}(a)+f(b)= \)

(d) \( \sqrt{1+x^{2}} \)

यदि \( g(x) = 2^{x} \), तो इसके इनवर्स फ़ंक्शन की पहचान करें और उसका डोमेन तथा रेंज लिखिए।

The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)).

I. Work Out (Show the necessary steps) For the rational functions (a-d) given below answer question (i-vi) each of the following i. Find the domain ii. \( \quad \) Find hole, if any iii. Find the vertical asymptote(s), if any iv. Find the horizontal or oblique asymptote, if any v. Find the \( x \)-intercept(s) and \( y \) - intercept(s) vi. \( \quad \) Prepare sign chart vii. Sketch the graph of each rational function a. \( \quad f(x)=\frac{x+2}{x^{2}-9} \)