Math  /  Algebra

Question(2 points) Let f(x)=3x2+3f(x)=3 x^{2}+3 and g(x)=2x2+8g(x)=2 x^{2}+8 Find and simplify each of the following functions: (a) (f+g)(x)=(f+g)(x)= (b) (fg)(x)=(f-g)(x)= (c) (fg)(x)=(f \cdot g)(x)= (d) (fg)(x)=\left(\frac{f}{g}\right)(x)= help (formulas)

Studdy Solution
To find (fg)(x)\left(\frac{f}{g}\right)(x), divide f(x) f(x) by g(x) g(x) :
(fg)(x)=3x2+32x2+8 \left(\frac{f}{g}\right)(x) = \frac{3x^2 + 3}{2x^2 + 8}
Simplify the fraction by factoring if possible. Here, factor out the greatest common factor from the numerator and the denominator:
Numerator: 3x2+3=3(x2+1) 3x^2 + 3 = 3(x^2 + 1)
Denominator: 2x2+8=2(x2+4) 2x^2 + 8 = 2(x^2 + 4)
Thus, the simplified form is:
(fg)(x)=3(x2+1)2(x2+4) \left(\frac{f}{g}\right)(x) = \frac{3(x^2 + 1)}{2(x^2 + 4)}

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