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Suppose that the functions \newlineff and \newlinegg are defined as follows.\newlinef(x)=9xg(x)=4x+3f(x)=\frac{9}{x}\quad g(x)=\frac{4}{x+3}\newlineFind \newlinegf\frac{g}{f}. Then, give its domain using an interval or union of intervals.\newlineSimplify your answers.\newline(gf)(x)=∏\left(\frac{g}{f}\right)(x)=\prod\newlineDomain of \newlinegf\frac{g}{f} :

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Q. Suppose that the functions \newlineff and \newlinegg are defined as follows.\newlinef(x)=9xg(x)=4x+3f(x)=\frac{9}{x}\quad g(x)=\frac{4}{x+3}\newlineFind \newlinegf\frac{g}{f}. Then, give its domain using an interval or union of intervals.\newlineSimplify your answers.\newline(gf)(x)=∏\left(\frac{g}{f}\right)(x)=\prod\newlineDomain of \newlinegf\frac{g}{f} :
  1. Write Functions: Write down the functions f(x)f(x) and g(x)g(x).f(x)=9xf(x) = \frac{9}{x} and g(x)=4x+3g(x) = \frac{4}{x+3}
  2. Find Quotient: Find the quotient (g/f)(x)(g/f)(x).(g/f)(x)=g(x)f(x)=4(x+3)9x(g/f)(x) = \frac{g(x)}{f(x)} = \frac{\frac{4}{(x+3)}}{\frac{9}{x}}
  3. Simplify Expression: Simplify the expression for (g/f)(x)(g/f)(x).(g/f)(x)=4(x+3)×x9=4x9(x+3)(g/f)(x) = \frac{4}{(x+3)} \times \frac{x}{9} = \frac{4x}{9(x+3)}
  4. Determine Domain: Simplify the expression further if possible.\newlineThe expression (4x9(x+3))(\frac{4x}{9(x+3)}) is already in its simplest form.
  5. Write Domain: Determine the domain of (g/f)(x)(g/f)(x).\newlineThe domain of (g/f)(x)(g/f)(x) is all real numbers except where the denominator is zero. So we set the denominator equal to zero and solve for xx.\newline9(x+3)=09(x+3) = 0\newlinex+3=0x+3 = 0\newlinex=−3x = -3
  6. Write Domain: Determine the domain of (g/f)(x)(g/f)(x). The domain of (g/f)(x)(g/f)(x) is all real numbers except where the denominator is zero. So we set the denominator equal to zero and solve for x. 9(x+3)=09(x+3) = 0 x+3=0x+3 = 0 x=−3x = -3 Write the domain using an interval or union of intervals. The domain of (g/f)(x)(g/f)(x) is all real numbers except x=−3x = -3. Therefore, the domain is (−∞,−3)∪(−3,∞)(-\infty, -3) \cup (-3, \infty).

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