The domain of f(x) is the set of all real values except 7.
The domain of g(x) is the set of all real values except -3.
In this question, we divide the two functions (g/f)(x) which will produce the new function, the new function is defined at any point of D(g)∩D(f) at which f(x)≠0.
Now,
Domain of (g/f)(x) is the set of all real values except 7 and -3, also exclude those points where f(x)=0.
e.g
f(x)=[tex] \sqrt{x} [/tex], g(x)=[tex] \sqrt{1-x} [/tex]
D(f)=[0, infinity) , D(g)= (-infinity, 1]
[tex] \frac{g}{f}(x)=\sqrt{\frac{1-x}{x}} [/tex] Domain=D(g) ∩ D(f)= (0, 1] (x=0 excluded, because f(x)=0 at this point)
Answer:
The domain of the function (g/f)(x) is : all the real numbers except zeros of f(x) and -3.
Step-by-step explanation:
We are given:
the domain of f(x) is the set of all real values except 7
and the domain of g(x) is the set of all real values except -3.
Now we are asked to find the domain of the function:
(g/f)(x).
We know that this function could also be written as:
[tex](g/f)(x)=\dfrac{g(x)}{f(x)}[/tex]
Hence this function is defined everywhere except the points where denominator or we can say function f(x) is zero and also the points where function g(x) is not defined.
Hence the domain of the function will depend on the domain of function g(x) and zeros of f(x).
Hence, the domain of the function (g/f)(x) is all the real numbers except zeros of f(x) and -3
