verify functions f and g are inverses of each other by showing that f(g(x)) = x and g(f(x)) = x.
f(x)= 5/(2 -x)
g (x) = 2 - 5/x
what's the problem?
f(x) = 5/(2-x) so
f(g) = 5/(2-g)
= 5/(2-[2 - 5/x])
= 5/(2 - 2 + 5/x)
= 5/(5/x)
= 5*x/5
= x
Do likewise for g(f)
To verify that functions f(x) and g(x) are inverses of each other, we need to show that f(g(x)) = x and g(f(x)) = x for all values of x.
Let's begin by evaluating f(g(x)):
f(g(x)) = f(2 - 5/x)
Here, we replace g(x) in f(x) with its expression (2 - 5/x). Now, simplify:
f(g(x)) = 5/(2 - (2 - 5/x))
= 5/(2 - 2 + 5/x)
= 5/(5/x)
= 5x/5
= x
Since f(g(x)) simplifies to x, we can conclude that f(x) and g(x) are inverses.
Now, let's evaluate g(f(x)):
g(f(x)) = g(5/(2 - x))
Similarly, we replace f(x) in g(x) with its expression (5/(2 - x)). Now, simplify:
g(f(x)) = 2 - 5/(5/(2 - x))
= 2 - 5 * (2 - x)/5
= 2 - (2 - x)
= 2 - 2 + x
= x
Therefore, g(f(x)) simplifies to x as well. This confirms that g(x) and f(x) are indeed inverses of each other.