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
please show work
Well, it seems like someone is having fun with functions! Let's dive into it:
First, let's find f(g(x)). To do that, we need to substitute g(x) into f(x): f(g(x)) = 9/(g(x)). Now let's replace g(x) with its definition: f(g(x)) = 9/(9/x) = 9x/9 = x. So, f(g(x)) = x.
As for the values that may be excluded, we need to consider the domain restrictions. Since both f(x) and g(x) have a denominator of x, we must exclude x = 0 from the domain. This means that x = 0 is not allowed for the compositions f(g(x)) and g(f(x)).
Now, let's find g(f(x)). To do that, we need to substitute f(x) into g(x): g(f(x)) = 9/(f(x)). Replacing f(x) with its definition, we get g(f(x)) = 9/(9/x) = 9x/9 = x. So, g(f(x)) = x as well.
Regarding the inverse functions, let's find the inverse of f(x) first. To do that, we swap x and y, and solve for y:
x = 9/y. Cross-multiplying, we get xy = 9. Dividing both sides by x, we find y = 9/x. Therefore, the inverse of f(x) is f^(-1)(x) = 9/x.
Now, let's find the inverse of g(x). Following the same steps, we have x = 9/y, and solving for y: y = 9/x. So, the inverse of g(x) is g^(-1)(x) = 9/x.
Voila! The compositions f(g(x)) and g(f(x)) both simplify to x. The excluded values are x = 0 due to the domain restrictions. And the inverse functions are f^(-1)(x) = 9/x and g^(-1)(x) = 9/x.
To find f(g(x)), we simply substitute g(x) into the function f(x).
Given g(x) = 9/x, we substitute x with 9/x in the function f(x) which is f(g(x)) = 9/(9/x).
To simplify the expression, we multiply the numerator by the reciprocal of the denominator:
f(g(x))= (9 * x)/(9) = x.
Therefore, f(g(x)) = x.
Now, to find g(f(x)), we substitute f(x) into the function g(x).
Given f(x) = 9/x, we substitute x with 9/x in the function g(x) which is g(f(x)) = 9/(9/x).
Again, we simplify the expression by multiplying the numerator by the reciprocal of the denominator:
g(f(x)) = (9 * x)/(9) = x.
Therefore, g(f(x)) = x.
The values that may be excluded in both cases are the values of x that make the denominators equal to zero since division by zero is undefined.
For f(g(x)), the denominator 9/x becomes zero when x = 0.
For g(f(x)), the denominator 9/(9/x) also becomes zero when x = 0.
Hence, the value x = 0 is excluded in both cases.