Given two functions as:
f(x) = x2-x-1 and g(x) = 3/x
Find fog(x) also find the domain of f, g and fog
domain of f : any real number
domain of g : any real number except x ≠ 0
fog
= f(g(x))
= f(3/x)
- (3/x)^2 - 3/x - 1
clearly the domain of fog is any real number, except x≠0
last line in calculation should say
= (3/x)^2 - 3/x - 1
To find fog(x), we need to substitute g(x) into f(x).
fog(x) = f(g(x))
First, let's find g(x) by substituting x into the function g(x) = 3/x:
g(x) = 3/x
Next, substitute g(x) back into f(x):
fog(x) = f(g(x)) = f(3/x)
Now, let's find the domain of f, g, and fog:
The domain of a function represents the set of all possible input values (x) for which the function is defined.
For f(x) = x^2 - x - 1, the domain is all real numbers because there are no restrictions on x.
For g(x) = 3/x, the function is undefined when x = 0 because division by zero is not allowed. Therefore, the domain of g(x) is all real numbers except x = 0.
Now, let's find the domain of fog(x) = f(g(x)):
Since g(x) = 3/x, we need to find the values of x that make g(x) defined. As mentioned before, g(x) is undefined when x = 0 because division by zero is not allowed.
Therefore, the domain of fog(x) is all real numbers except x = 0.
In summary:
- The domain of f(x) = x^2 - x - 1 is all real numbers.
- The domain of g(x) = 3/x is all real numbers except x = 0.
- The domain of fog(x) = f(g(x)) is all real numbers except x = 0.