For the function f(x) = (x - 2) / (x^2 + 1), find the critical properties necessary and sketch the function. Describe each special case, and state the domain of the function.

Steps to Solve

1. Identify the Function and Domain The function is given as: $$ f(x) = \frac{x - 2}{x^2 + 1} $$ Since the denominator $x^2 + 1$ is always positive (no values of $x$ make it zero), the domain is all real numbers: $$ \text{Domain: } (-\infty, \infty) $$

2. Find Critical Points To find critical points, we need to calculate the first derivative and set it equal to zero.

First, apply the quotient rule, where: $$ f'(x) = \frac{(g(x) \cdot f'(x) - f(x) \cdot g'(x))}{(g(x))^2} $$

Here, ( f(x) = x - 2 ) and ( g(x) = x^2 + 1 ).

Calculating derivatives: $$ f'(x) = 1 $$ $$ g'(x) = 2x $$

Now applying the quotient rule: $$ f'(x) = \frac{(x^2 + 1)(1) - (x - 2)(2x)}{(x^2 + 1)^2} $$ Simplifying: $$ f'(x) = \frac{x^2 + 1 - 2x^2 + 4x}{(x^2 + 1)^2} $$ $$ = \frac{-x^2 + 4x + 1}{(x^2 + 1)^2} $$

Setting the numerator equal to zero: $$ -x^2 + 4x + 1 = 0 $$

3. Solve the Quadratic Equation Using the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ): Here, ( a = -1 ), ( b = 4 ), and ( c = 1 ).

Calculating the discriminant: $$ D = 4^2 - 4 \cdot (-1) \cdot 1 = 16 + 4 = 20 $$

Thus, the roots are: $$ x = \frac{-4 \pm \sqrt{20}}{2(-1)} = \frac{-4 \pm 2\sqrt{5}}{-2} $$ $$ = 2 \mp \sqrt{5} $$

4. Evaluate the Second Derivative (Optional) To determine if the critical points are maxima or minima, you could compute the second derivative.

5. Sketch the Function At the critical points ( x = 2 - \sqrt{5} ) and ( x = 2 + \sqrt{5} ), evaluate $f(x)$ to see the function behavior:

6. Special Cases Determine y-intercept: $$ f(0) = -2 $$

As ( x \to \pm \infty ), the function approaches 0 since degree of numerator is less than that of denominator.