consider the function f(x)=9/x and g(x)=9/x
find f(g)x))and any values that may be excluded
what is g(f(x)) and any values that may be excluded
and functions f and g inverse of eachother
To find f(g(x)), we need to substitute g(x) into the function f(x).
Given that g(x) = 9/x, we can substitute this into f(x):
f(g(x)) = f(9/x)
Now, f(x) = 9/x, so:
f(9/x) = 9/(9/x)
To simplify this, we can multiply the numerator and denominator of the fraction by x:
f(9/x) = (9*x) / 9
Canceling out the 9 in the numerator and denominator:
f(9/x) = x
Therefore, f(g(x)) simplifies to x.
Now, let's find g(f(x)). We need to substitute f(x) into g(x):
Given that f(x) = 9/x, we can substitute this into g(x):
g(f(x)) = g(9/x)
Now, g(x) = 9/x, so:
g(9/x) = 9/(9/x)
Again, to simplify this, we can multiply the numerator and denominator of the fraction by x:
g(9/x) = (9*x) / 9
Canceling out the 9 in the numerator and denominator:
g(9/x) = x
Therefore, g(f(x)) simplifies to x.
As for the values that may be excluded, in both f(g(x)) and g(f(x)), the only value that cannot be used is x=0. This is because division by zero is undefined in mathematics. For all other values of x, both f(g(x)) and g(f(x)) are equal to x.