Question

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. Which of the following describes the domain of (g o f) (x)?

all real values except x not-equals negative 3 and the x for which f (x) not-equals 7
all real values except x not-equals negative 3 and the x for which f (x) not-equals negative 3
all real values except x not-equals 7 and the x for which f (x) not-equals 7
all real values except x not-equals 7 and the x for which f (x) not-equals negative 3

Answer 1

The last is the correct option

"all real values except x not-equals 7 and the x for which f (x) not-equals negative 3"

Explanation:

Domain and Range of Functions

Given the function f(x), the domain of f is the set of all the values that x can take such f(x) exists. The range of f is the set of all the values that f takes.

We have a problem where we have to find the domain of a composite function. Let's recall that being f and g real functions, then

`g\circ f=g(f(x))`

is the composite function of f and g.

We know the domain of f is the set of all real values except 7, and the domain of g is the set of all real values except –3.

Since f is the innermost function, the domain of the composite function is directly restricted by the domain of f. So, x cannot be 7.

Now, g takes f as its independent variable, and we know the domain of g excludes -3. It can be found that f(x) cannot be -3 because it will cause g not to exist.

Thus, the domain of
`g\circ f` is

All real numbers except x=7 and those where f(x)=-3

The last is the correct option