Let f(x)=x^2+9x+20 and g(x)=x^3-4x Find (fg)(x) and ((f)/(g))(x) (fg)(x)=vert x^5+5x^4-16x^2-80x ((f)/(g))(x)=vert (x+4)/(x-2)x State the domain of each. Domain of (fg)(x) all real numbers Domain of ((f)/(g))(x) xvert xneq 0,2 : all real numbers xvert xneq 0 xvert xneq -2,0,2 xvert xneq 0,2 Evaluate the following. (fg)(-1)=66x ((f)/(g))(-1)=vert -1x

To solve the given problem, we need to find \((fg)(x)\) and \(\left(\frac{f}{g}\right)(x)\), and then determine their domains.<br /><br />1. **Finding \((fg)(x)\):**<br /><br /> Given:<br /> \[<br /> f(x) = x^2 + 9x + 20<br /> \]<br /> \[<br /> g(x) = x^3 - 4x<br /> \]<br /><br /> The product \((fg)(x)\) is:<br /> \[<br /> (fg)(x) = f(x) \cdot g(x) = (x^2 + 9x + 20)(x^3 - 4x)<br /> \]<br /><br /> Expanding this product:<br /> \[<br /> = x^2(x^3 - 4x) + 9x(x^3 - 4x) + 20(x^3 - 4x)<br /> \]<br /> \[<br /> = x^5 - 4x^3 + 9x^4 - 36x^2 + 20x^3 - 80x<br /> \]<br /> \[<br /> = x^5 + 9x^4 + (20x^3 - 4x^3) - 36x^2 - 80x<br /> \]<br /> \[<br /> = x^5 + 9x^4 + 16x^3 - 36x^2 - 80x<br /> \]<br /><br /> So, \((fg)(x) = x^5 + 9x^4 + 16x^3 - 36x^2 - 80x\).<br /><br />2. **Finding \(\left(\frac{f}{g}\right)(x)\):**<br /><br /> The quotient \(\left(\frac{f}{g}\right)(x)\) is:<br /> \[<br /> \left(\frac{f}{g}\right)(x) = \frac{x^2 + 9x + 20}{x^3 - 4x}<br /> \]<br /><br /> To simplify, factor both the numerator and the denominator:<br /><br /> - Numerator: \(x^2 + 9x + 20 = (x + 4)(x + 5)\)<br /> - Denominator: \(x^3 - 4x = x(x^2 - 4) = x(x - 2)(x + 2)\)<br /><br /> Thus:<br /> \[<br /> \left(\frac{f}{g}\right)(x) = \frac{(x + 4)(x + 5)}{x(x - 2)(x + 2)}<br /> \]<br /><br />3. **Domain of \((fg)(x)\):**<br /><br /> Since \((fg)(x)\) is a polynomial, its domain is all real numbers.<br /><br />4. **Domain of \(\left(\frac{f}{g}\right)(x)\):**<br /><br /> The domain excludes values that make the denominator zero:<br /> \[<br /> x(x - 2)(x + 2) = 0 \implies x = 0, 2, -2<br /> \]<br /><br /> Therefore, the domain is \(\{ x \mid x \neq -2, 0, 2 \}\).<br /><br />5. **Evaluate \((fg)(-1)\):**<br /><br /> Substitute \(x = -1\) into \((fg)(x)\):<br /> \[<br /> (fg)(-1) = (-1)^5 + 9(-1)^4 + 16(-1)^3 - 36(-1)^2 - 80(-1)<br /> \]<br /> \[<br /> = -1 + 9 - 16 - 36 + 80<br /> \]<br /> \[<br /> = 36<br /> \]<br /><br />6. **Evaluate \(\left(\frac{f}{g}\right)(-1)\):**<br /><br /> Substitute \(x = -1\) into \(\left(\frac{f}{g}\right)(x)\):<br /> \[<br /> \left(\frac{f}{g}\right)(-1) = \frac{(-1)^2 + 9(-1) + 20}{(-1)^3 - 4(-1)}<br /> \]<br /> \[<br /> = \frac{1 - 9 + 20}{-1 + 4}<br /> \]<br /> \[<br /> = \frac{12}{3} = 4<br /> \]<br /><br />In summary:<br />- \((fg)(x) = x^5 + 9x^4 + 16x^3 - 36x^2 - 80x\)<br />- Domain of \((fg)(x)\): all real numbers<br />- \(\left(\frac{f}{g}\right)(x) = \frac{(x + 4)(x + 5)}{x(x - 2)(x + 2)}\)<br />- Domain of \(\left(\frac{f}{g}\right)(x)\): \(\{ x \mid x \neq -2, 0, 2 \}\)<br />- \((fg)(-1) = 36\)<br />- \(\left(\frac{f}{g}\right)(-1) = 4\)