Let f(x)=x^2-2x-3 and g(x)=x^2-4 .Find (fg)(x) and ((f)/(g))(x) (fg)(x)=square ((f)/(g))(x)=square State the domain of each. Domain of (fg)(x) square Domain of ((f)/(g))(x) square Evaluate the following. (fg)(-3)=square ((f)/(g))(-3)= square : all real numbers xvert xneq 0 xvert xneq -2,2 xvert xneq 2

To find \((fg)(x)\) and \(\left(\frac{f}{g}\right)(x)\), we need to perform the following operations:<br /><br />1. **Find \((fg)(x)\):**<br /><br /> \[<br /> (fg)(x) = f(x) \cdot g(x) = (x^2 - 2x - 3)(x^2 - 4)<br /> \]<br /><br /> To multiply these polynomials, use the distributive property (also known as FOIL for binomials):<br /><br /> \[<br /> (x^2 - 2x - 3)(x^2 - 4) = x^2(x^2 - 4) - 2x(x^2 - 4) - 3(x^2 - 4)<br /> \]<br /><br /> Calculate each term:<br /><br /> \[<br /> x^2(x^2 - 4) = x^4 - 4x^2<br /> \]<br /> \[<br /> -2x(x^2 - 4) = -2x^3 + 8x<br /> \]<br /> \[<br /> -3(x^2 - 4) = -3x^2 + 12<br /> \]<br /><br /> Combine all terms:<br /><br /> \[<br /> x^4 - 4x^2 - 2x^3 + 8x - 3x^2 + 12<br /> \]<br /><br /> Simplify by combining like terms:<br /><br /> \[<br /> x^4 - 2x^3 - 7x^2 + 8x + 12<br /> \]<br /><br /> So, \((fg)(x) = x^4 - 2x^3 - 7x^2 + 8x + 12\).<br /><br />2. **Find \(\left(\frac{f}{g}\right)(x)\):**<br /><br /> \[<br /> \left(\frac{f}{g}\right)(x) = \frac{x^2 - 2x - 3}{x^2 - 4}<br /> \]<br /><br /> The expression is already simplified.<br /><br />3. **State the domain of each:**<br /><br /> - The domain of \((fg)(x)\) is all real numbers because it is a polynomial function. Therefore, the domain is:<br /> \[<br /> \text{Domain of } (fg)(x) = \text{all real numbers}<br /> \]<br /><br /> - The domain of \(\left(\frac{f}{g}\right)(x)\) excludes values that make the denominator zero. Solve \(x^2 - 4 = 0\):<br /><br /> \[<br /> x^2 - 4 = (x - 2)(x + 2) = 0<br /> \]<br /><br /> This gives \(x = 2\) and \(x = -2\). Therefore, the domain is:<br /> \[<br /> \text{Domain of } \left(\frac{f}{g}\right)(x) = \{ x \mid x \neq -2, 2 \}<br /> \]<br /><br />4. **Evaluate the following:**<br /><br /> - \((fg)(-3)\):<br /><br /> Substitute \(x = -3\) into \((fg)(x)\):<br /><br /> \[<br /> (fg)(-3) = (-3)^4 - 2(-3)^3 - 7(-3)^2 + 8(-3) + 12<br /> \]<br /><br /> Calculate each term:<br /><br /> \[<br /> (-3)^4 = 81, \quad -2(-3)^3 = 54, \quad -7(-3)^2 = -63, \quad 8(-3) = -24<br /> \]<br /><br /> Add them up:<br /><br /> \[<br /> 81 + 54 - 63 - 24 + 12 = 60<br /> \]<br /><br /> So, \((fg)(-3) = 60\).<br /><br /> - \(\left(\frac{f}{g}\right)(-3)\):<br /><br /> Substitute \(x = -3\) into \(\left(\frac{f}{g}\right)(x)\):<br /><br /> \[<br /> \left(\frac{f}{g}\right)(-3) = \frac{(-3)^2 - 2(-3) - 3}{(-3)^2 - 4}<br /> \]<br /><br /> Calculate the numerator and denominator:<br /><br /> \[<br /> \text{Numerator: } (-3)^2 - 2(-3) - 3 = 9 + 6 - 3 = 12<br /> \]<br /> \[<br /> \text{Denominator: } (-3)^2 - 4 = 9 - 4 = 5<br /> \]<br /><br /> Therefore:<br /><br /> \[<br /> \left(\frac{f}{g}\right)(-3) = \frac{12}{5}<br /> \]<br /><br />In summary:<br /><br />- \((fg)(x) = x^4 - 2x^3 - 7x^2 + 8x + 12\)<br />- \(\left(\frac{f}{g}\right)(x) = \frac{x^2 - 2x - 3}{x^2 - 4}\)<br />- Domain of \((fg)(x)\): all real numbers<br />- Domain of \(\left(\frac{f}{g}\right)(x)\): \(\{ x \mid x \neq -2, 2 \}\)<br />- \((fg)(-3) = 60\)<br />- \(\left(\frac{f}{g}\right)(-3) = \frac{12}{5}\)