# If f(x)=x^2+5x+6 and g(x)=3x+1, what is the domain of f(x)/g(x)?

jeew-m | Certified Educator

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`f(x)=x^2+5x+6`

`g(x)=3x+1`

`(f(x))/(g(x)) = (x^2+5x+6)/(3x+1)`

This function `(f(x))/(g(x))` will be undefinable when `(3x+1)=0` OR `x = -1/3` .

For every other x there is no problem with `(f(x))/(g(x))` .

So the domain of `(f(x))/(g(x))` is;

`x in R-[-1/3]`

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tonys538 | Student

The domain of a function y = f(x) is all the values that x can take for which y = f(x) is real and defined.

For f(x)=x^2+5x+6 and g(x)=3x+1, the value of y = f(x)/g(x) = (x^2 + 5x +6)/(3x+1)

= (x^2 + 2x + 3x + 6)/(3x+1)

= (x(x + 2) + 3(x + 2))/(3x+1)

= ((x+3)(x+2))/(3x+1)

The value of the numerator is defined for all values of x, the expression (f(x))/(g(x)) is not defined if g(x) = 0

3x+1 = 0

x = -1/3

The domain of (f(x))/(g(x)) is the set of real values except x = -1/3

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