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What is the domain of (fg)(x)\left(\frac{f}{g}\right)(x)?\newlinef(x)=2xf(x) = 2x\newlineg(x)=5x+3g(x) = -5x + 3\newlineChoices:\newline[A]All real numbers[\text{A]All real numbers}\newline[B]All real numbers except 35[\text{B]All real numbers except } \frac{3}{5}\newline[C]All real numbers except 35[\text{C]All real numbers except } -\frac{3}{5}\newline[D]All real numbers except 53[\text{D]All real numbers except } \frac{5}{3}

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Q. What is the domain of (fg)(x)\left(\frac{f}{g}\right)(x)?\newlinef(x)=2xf(x) = 2x\newlineg(x)=5x+3g(x) = -5x + 3\newlineChoices:\newline[A]All real numbers[\text{A]All real numbers}\newline[B]All real numbers except 35[\text{B]All real numbers except } \frac{3}{5}\newline[C]All real numbers except 35[\text{C]All real numbers except } -\frac{3}{5}\newline[D]All real numbers except 53[\text{D]All real numbers except } \frac{5}{3}
  1. Identify Formula: Identify the formula for ((f)/(g))(x)((f)/(g))(x).((f)/(g))(x)((f)/(g))(x) is the quotient of f(x)f(x) and g(x)g(x), which means we need to divide f(x)f(x) by g(x)g(x).((f)/(g))(x)=f(x)/g(x)((f)/(g))(x) = f(x) / g(x)
  2. Given Functions: We have the functions:\newlinef(x)=2xf(x) = 2x\newlineg(x)=5x+3g(x) = -5x + 3\newlineTo find the domain of (fg)(x)\left(\frac{f}{g}\right)(x), we need to determine the values of xx for which g(x)g(x) is not equal to 00, since division by zero is undefined.
  3. Find Domain: Solve for xx when g(x)=0g(x) = 0 to find the excluded value from the domain.\newlineg(x)=0g(x) = 0 \newline5x+3=0–5x + 3 = 0 \newline5x=3–5x = –3 \newlinex=3/(5)x = –3 / (–5) \newlinex=35x = \frac{3}{5}
  4. Solve for Excluded Value: We found that g(x)=0g(x) = 0 when x=35x = \frac{3}{5}. This means that x=35x = \frac{3}{5} is the value that cannot be included in the domain of (fg)(x)\left(\frac{f}{g}\right)(x) because it would make the denominator zero.

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