Rational Function 4A2Daa

1. **State the problem:** We need to construct a rational function $f(x) = \frac{P(x)}{Q(x)}$ that satisfies: - Vertical asymptotes at $x = -3$ and $x = 1$. - A hole at $x = -2$. - x-intercepts at $x = 4$ and $x = 0$. - Horizontal asymptote at $y = 2$. 2. **Recall formulas and rules:** - Vertical asymptotes occur where the denominator is zero but the numerator is not zero. - Holes occur where both numerator and denominator are zero at the same $x$ value. - x-intercepts occur where the numerator is zero but the denominator is not zero. - Horizontal asymptote depends on degrees of numerator and denominator: - If degrees are equal, horizontal asymptote is ratio of leading coefficients. 3. **Construct denominator $Q(x)$:** - Vertical asymptotes at $x = -3$ and $x = 1$ imply factors $(x + 3)$ and $(x - 1)$ in denominator. - Hole at $x = -2$ implies factor $(x + 2)$ in denominator and numerator. So, $$Q(x) = (x + 3)(x - 1)(x + 2)$$ 4. **Construct numerator $P(x)$:** - Hole at $x = -2$ means numerator also has factor $(x + 2)$. - x-intercepts at $x = 4$ and $x = 0$ imply factors $(x - 4)$ and $x$ in numerator. So, $$P(x) = k \cdot x (x - 4)(x + 2)$$ where $k$ is a constant to be determined. 5. **Determine $k$ using horizontal asymptote:** - Degree numerator = degree denominator = 3. - Horizontal asymptote $y = 2$ means leading coefficient ratio = 2. Leading coefficient numerator: $k \cdot 1 \cdot 1 \cdot 1 = k$ Leading coefficient denominator: $1 \cdot 1 \cdot 1 = 1$ So, $$\frac{k}{1} = 2 \implies k = 2$$ 6. **Final function:** $$f(x) = \frac{2x(x - 4)(x + 2)}{(x + 3)(x - 1)(x + 2)}$$ 7. **Simplify by canceling hole factor:** $$f(x) = \frac{\cancel{(x + 2)} 2x (x - 4)}{(x + 3)(x - 1) \cancel{(x + 2)}} = \frac{2x(x - 4)}{(x + 3)(x - 1)}$$ 8. **Verify conditions:** - Vertical asymptotes at $x = -3$ and $x = 1$ because denominator zero and numerator nonzero. - Hole at $x = -2$ because factor canceled. - x-intercepts at $x = 0$ and $x = 4$ because numerator zero and denominator nonzero. - Horizontal asymptote $y = 2$ because degrees equal and leading coefficient ratio is 2. **Answer:** $$\boxed{f(x) = \frac{2x(x - 4)}{(x + 3)(x - 1)}}$$