91Ó°ÊÓ

The domain of a function is the set of all possible input values. The function is not defined when the denominator is equal to zero. Set the denominator equal to zero and solve for x: $$ x^{2}-1=0 \implies x^2=1 \implies x=\pm1 $$ The function is not defined at \(x= \pm1\). Therefore, the domain is: $$ x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty) $$

To find the x-intercept(s), set the function equal to zero and solve for x: $$ f(x) = \frac{3x-5}{ x^2-1} = 0 \implies 3x-5 = 0 \implies x=\frac{5}{3} $$ So there is one x-intercept at \(x = \frac{5}{3}\). To find the y-intercept, evaluate f(0): $$ f(0) = \frac{3(0)-5}{0^2-1} = \frac{-5}{-1} = 5 $$ So, the y-intercept is at \(y=5\).

Check if the function is even, odd, or neither. \(f(-x) = \frac{3(-x) - 5}{(-x)^2 - 1} = \frac{-3x - 5}{x^2 - 1}\). Since \(f(-x) \neq f(x)\) and \(f(-x) \neq -f(x)\), the function is neither even nor odd function.

Analyze the vertical asymptotes by finding the limit as x approaches the discontinuities \(\pm 1\): $$ \lim_{x\to-1}f(x) = \pm\infty, \\ \lim_{x\to 1}f(x) = \pm\infty, $$ There are vertical asymptotes at \(x= -1\) and \(x=1\). Analyze the horizontal asymptotes by finding the limit as x approaches \(\pm\infty\): $$ \lim_{x\to\pm\infty}f(x) = \lim_{x\to\pm\infty}\frac{3x - 5}{x^2 - 1} = 0 $$ There is a horizontal asymptote at \(y=0\).

Find the first and second derivatives of the function: $$ f'(x) = \frac{(3x^2 - 3)-(3x-5)(2x)}{(x^2-1)^2} \\ = \frac{-6x^2 + 10x + 3}{(x^2-1)^2} $$ $$ f''(x) = \frac{\left(-12x^2 + 20x + 6\right)\left(x^2 - 1\right)^2 - 2\left(x^2 - 1\right)^3}{(x^2 - 1)^4} =\frac{-12x^4+36x^2-40x^3+20x^2+4}{(x^2 - 1)^3} $$ To find possible locations of extreme values, we locate the critical points by setting the first derivative equal to zero and solving for x. However, since the function is neither even nor odd, it has unpredictable symmetry and we can't definitively say the local extreme values are at the critical points. To find inflection points, set \(f''(x)\) equal to zero and solve for \(x\). Since solving for \(x\) in this case is highly complex, it is practical to use a graphing utility to locate the inflection points. From the above steps, plot the graph considering domain, intercepts, asymptotes, and inflection points. The complete graph will depict the behavior of the function, and can be used to analyze the relationships between these important features.