Determine whether f(x)=5x+1/x and g(x)= x/5x+1 are inverse functions. Explain how you know.
see the post just below yours.
To determine whether two functions are inverse functions, we need to check if they undo each other when composed. In other words, if we substitute one function into the other and then simplify, we should end up with the input value.
To check if f(x) and g(x) are inverse functions, we will substitute f(g(x)) into g(x) and simplify.
Let's start with f(x) = 5x + 1/x and substitute g(x) into f(x):
f(g(x)) = 5(g(x)) + 1/g(x)
Now substitute g(x) = x/(5x + 1):
f(g(x)) = 5(x/(5x + 1)) + 1/(x/(5x + 1))
Simplifying the expression:
f(g(x)) = (5x)/(5x + 1) + (5x + 1)/x
To combine the fractions, we need a common denominator, which will be x(5x + 1):
f(g(x)) = (5x^2 + x + (5x + 1)(5x + 1))/(x(5x + 1))
Expanding and simplifying:
f(g(x)) = (5x^2 + x + 25x^2 + 10x + 1)/(x(5x + 1))
= (30x^2 + 11x + 1)/(x(5x + 1))
Now let's simplify g(f(x)) by substituting f(x) into g(x):
g(f(x)) = f(x)/(5f(x) + 1)
= (5x + 1/x)/(5(5x + 1) + 1)
= (5x + 1/x)/(25x + 6)
To combine the fractions, we need a common denominator, which will be x:
g(f(x)) = ((5x^2 + 1)/(x^2))/(25x + 6)
Now we can clearly see the expressions for f(g(x)) and g(f(x)).
Since f(g(x)) = (30x^2 + 11x + 1)/(x(5x + 1)) and g(f(x)) = ((5x^2 + 1)/(x^2))/(25x + 6), we can conclude that f(x) and g(x) are **not** inverse functions.