Given a real parameter $a$, with $a \neq 0$, consider the function $f_{a}$ defined as follows: $$f_{a}(x) = \frac{x^{2} - ax}{x^{2} - a}$$ whose graph will be denoted by $\Omega_{a}$. a) As the parameter $a$ varies, determine the domain of $f_{a}$, study any discontinuities and write the equations of all its asymptotes. b) Show that, for $a \neq 1$, all graphs $\Omega_{a}$ intersect their horizontal asymptote at the same point and share the same tangent line at the origin. c) As $a < 1$ varies, identify the intervals of monotonicity of the function $f_{a}$. Study the function $f_{-1}(x)$ and sketch its graph $\Omega_{-1}$. d) Determine the area of the bounded region between the graph $\Omega_{-1}$, the line tangent to it at the origin and the line $x = \sqrt{3}$.
Given a real parameter $a$, with $a \neq 0$, consider the function $f_{a}$ defined as follows:
$$f_{a}(x) = \frac{x^{2} - ax}{x^{2} - a}$$
whose graph will be denoted by $\Omega_{a}$.
a) As the parameter $a$ varies, determine the domain of $f_{a}$, study any discontinuities and write the equations of all its asymptotes.
b) Show that, for $a \neq 1$, all graphs $\Omega_{a}$ intersect their horizontal asymptote at the same point and share the same tangent line at the origin.
c) As $a < 1$ varies, identify the intervals of monotonicity of the function $f_{a}$. Study the function $f_{-1}(x)$ and sketch its graph $\Omega_{-1}$.
d) Determine the area of the bounded region between the graph $\Omega_{-1}$, the line tangent to it at the origin and the line $x = \sqrt{3}$.