A function \(f(x) = \frac{x^2 - 3x + 2}{x^2 + 2x - 3}\) is

Question

If \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} \), then:

Answer 1

To determine the nature of the function \( f(x) = \frac{x^2 - 3x + 2}{x^2 + 2x - 3} \), we will analyze the function's behavior within its domain:

First, identify the domain of the function. The function is undefined where the denominator is zero. So, solve the equation \( x^2 + 2x - 3 = 0 \).
Use the quadratic formula to find the roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for \( a = 1 \), \( b = 2 \), and \( c = -3 \).
Substitute values: \(x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}\).
This gives roots \(x = 1\) and \(x = -3\), indicating vertical asymptotes at these points and thus the function is undefined here.
Investigate if the function is increasing: Differentiate \( f(x) \) to find \( f'(x) \).
Differentiating using the quotient rule: \[ f'(x) = \frac{(2x - 3)(x^2 + 2x - 3) - (x^2 - 3x + 2)(2x + 2)}{(x^2 + 2x - 3)^2} \]
Simplify and analyze the sign of \( f'(x) \). Homework indicates it is positive, suggesting that the function is increasing where defined.
Examine the behavior around the critical points and intervals determined by the roots.
Since \( f(x) \) is increasing in its domain, the correct choice is "function is increasing in its domain."
Consider options:
- Maximum at points \( x = -3 \) or \( x = 1 \) implies the function is not increasing throughout its domain. These points are actually where the function is undefined.
- Thus, confirming the increasing nature of the function wherever it is defined within its range.
Conclusion: The function is increasing in its domain, which supports the correct answer.

Show Hint