Determine whether f(x)=5x+1/x and g(x)= x/5x+1 are inverse functions. Explain how you know.
Help please?
To determine whether two functions, f(x) and g(x), are inverse functions, we need to follow two steps:
1. Find the composition of the functions, f(g(x)), and simplify it.
2. Determine whether the simplified composition equals the identity function, f(g(x)) = x.
Let's start with the composition of the functions:
f(g(x)) = f(x/5x+1)
To evaluate this, we substitute g(x) = x/5x+1 into f(x):
f(g(x)) = 5 * (x/5x+1) + 1/(x/5x+1)
Now, let's simplify f(g(x)):
f(g(x)) = 5x/(5x+1) + 1/(x/5x+1)
Combining the fractions:
f(g(x)) = (5x^2 + 1)/(x(5x+1))
Now, to determine whether f(g(x)) equals x (the identity function), we simplify f(g(x)) as much as possible:
f(g(x)) = (5x^2 + 1)/(x(5x+1))
Since the numerator and denominator have different degrees, the functions f(x) and g(x) are not inverse functions.
Therefore, f(x)= 5x+1/x and g(x)= x/5x+1 are not inverse functions.
To determine whether two functions f(x) and g(x) are inverse functions, we need to check if the composition of the two functions will result in the input x. In other words, we need to verify if f(g(x)) = x and g(f(x)) = x.
Let's start by finding f(g(x)):
Replace g(x) in the function f(x) with x/5x+1:
f(g(x)) = 5(g(x)) + 1/(g(x))
Replace g(x) with x/5x+1:
f(g(x)) = 5(x/5x+1) + 1/(x/5x+1)
Simplify the expression:
f(g(x)) = 5x/5x+1 + 1/(x/5x+1)
To continue, we need a common denominator for the two fractions. We can multiply the first fraction by (5x+1) and the second fraction by 5x to get a common denominator.
f(g(x)) = (5x(5x+1))/[(5x+1)(5x+1)] + (5x)/[(x)(5x+1)]
Multiply the numerators:
f(g(x)) = (25x^2 + 5x)/(25x^2 + 10x + 1) + (5x)/(5x^2 + x)
To simplify the expression, we need to find a common denominator, which is (25x^2 + 10x + 1)(x).
f(g(x)) = [(25x^2 + 5x)(x)]/[(25x^2 + 10x + 1)(x)] + (5x(25x^2 + 10x + 1))/[(5x^2 + x)(25x^2 + 10x + 1)]
Combining the numerators:
f(g(x)) = (25x^3 + 5x^2 + 5x^2 + x)/(25x^3 + 10x^2 + x + 5x^3 + 2x + 1)
Simplifying further:
f(g(x)) = (25x^3 + 10x^2 + x)/(30x^3 + 3x + 1)
Now, let's find g(f(x)):
Replace f(x) in the function g(x) with 5x + 1/x:
g(f(x)) = f(x)/(5(f(x)) + 1)
Replace f(x) with 5x + 1/x:
g(f(x)) = (5x + 1/x)/[5(5x + 1/x) + 1]
Simplify the expression:
g(f(x)) = (5x + 1/x)/[(25x + 1/x) + 1]
To simplify further, we need to find a common denominator, which is (25x + 1/x)(x).
g(f(x)) = [(5x^2 + 1)/(x)]/[(x)(25x + 1/x) + (x)]
Simplifying the denominator:
g(f(x)) = (5x^2 + 1)/(25x^2 + 1 + x^2)
g(f(x)) = (5x^2 + 1)/(26x^2 + 1)
Now, we can compare f(g(x)) with x and g(f(x)) with x to determine if they are equal:
f(g(x)) = (25x^3 + 10x^2 + x)/(30x^3 + 3x + 1)
g(f(x)) = (5x^2 + 1)/(26x^2 + 1)
The two compositions f(g(x)) and g(f(x)) are not equal to x. Therefore, the functions f(x) = 5x + 1/x and g(x) = x/(5x + 1) are not inverse functions.
if f and g are inverses, then f(g(x) = g(f(x)) = x
I'll check f(g). You can do g(f)
Hmmm. I assume you meant
f(x) = (5x+1)/x and g(x) = x/(5x+1)
If so, it is immediately clear that g(x) = 1/f(x), not f-1(x). They are reciprocals, not inverses.
But, who knows? Maybe they are also inverses.
f(g) = (5g+1)/g = (5*x/(5x+1) + 1)/(x/(5x+1))
= (5x+5x+1)/x = (10x+1)/x = 10 + 1/x ≠ x
In fact, f-1(x) = 1/(x-5)
If we use f and g as you typed them, the situation is even worse, but they are clearly not inverses.