verify the functions are inverses of each other by showing that f(g(x)) =x show work.
f(x)=6/1-x g(x)= 1- 6/x
I will assume f(x) = 6/(1-x)
f(g(x))
= f(1 - 6/x)
= 6/(1 - (1 - 6/x)
= 6/(1-1+6/x)
= 6/(6/x)
= 6(x/6)
= x
these are hard to do - thank you
To verify that two functions, f(x) and g(x), are inverses of each other, we need to show that when we apply one function to the output of the other function, we get back the original input value.
In this case, we have:
f(x) = 6/(1 - x)
g(x) = 1 - 6/x
To show that f(g(x)) = x, we need to find the composition of these functions and simplify it:
Step 1: Find f(g(x)):
Substitute g(x) into f(x):
f(g(x)) = 6/(1 - g(x))
Replace g(x) with its expression:
f(g(x)) = 6/(1 - (1 - 6/x))
Simplify the inner expression:
f(g(x)) = 6/(1 - 1 + 6/x)
f(g(x)) = 6/(6/x)
Simplify further by multiplying the numerator by x and the denominator by 6:
f(g(x)) = (6 * x) / 6
Simplify:
f(g(x)) = x
Step 2: Since f(g(x)) = x, we have shown that f(x) and g(x) are inverses of each other.
To summarize, by substituting g(x) into f(x) and simplifying, we found that f(g(x)) simplifies to x. This confirms that the functions f(x) = 6/(1 - x) and g(x) = 1 - 6/x are inverses of each other.