[tex](\frac{g}{f})(3)=\frac{-13}{21}\\[/tex]
2/3 will not be included in the domain of g/f
Step-by-step explanation:
Given functions are:
[tex]g(x) =-2x^2+5\\f(x)=9x-6[/tex]
We have to calculate (f/g)(x) first
The steps will be as follows:
[tex](\frac{g}{f})(x)=\frac{g(x)}{f(x)}[/tex]
Putting the values of functions
[tex](\frac{g}{f})(x)=\frac{-2x^2+5}{9x-6}\\We\ have\ to\ find\ the\ value\ of\ (\frac{g}{f})(x)\ at\ 3\\So,\\(\frac{g}{f})(3)=\frac{-2(3)^2+5}{9(3)-6}\\(\frac{g}{f})(3)=\frac{-2(9)+5}{27-6}\\(\frac{g}{f})(3)=\frac{-18+5}{27-6}\\(\frac{g}{f})(3)=\frac{-13}{21}[/tex]
Domain of g/f:
[tex](\frac{g}{f})(x)=\frac{-2x^2+5}{9x-6}[/tex]
The function will be undefined if the denominator is zero.
To find the domain we will put the denominator equal to zero
So,
[tex]9x-6=0\\Adding\ 6\ on\ both\ sides\\9x-6+6=0+6\\9x=6\\Dividing\ both\ sides\ by\ 9\\\frac{9x}{9}=\frac{6}{9}\\x=\frac{2}{3}[/tex]
Hence, the function will be undefined on x=2/3 so 2/3 will not be included in the domain of (f/g)(x)
Answer:
[tex](\frac{g}{f})(3)=\frac{-13}{21}\\[/tex]
2/3 will not be included in the domain of g/f
Keywords: Domain, Operations on Functions
Learn more about functions at:
- brainly.com/question/4767370
- brainly.com/question/4770453
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