Question

Given that any function of f(x)=x+a/x, where a and can be any value,will have the inverse of the form f^-1(x)=a/x-1

find the inverse of f(x)=x+4/x
please add your working

Answer 1

To find the inverse of the function f(x) = x + 4/x, swap x and y, eliminate fractions, and solve the resulting equation using the quadratic formula, taking only the positive root to obtain the inverse function.

Step-by-step explanation:

To find the inverse of the function f(x) = x + 4/x, we want to solve for x in terms of y (we let y = f(x)). Here are the steps for finding the inverse:

1. Swap x and y in the equation: y = x + 4/x
2. Multiply both sides by x to get rid of the denominator: x*y = x2 + 4
3. Rearrange the equation to form a quadratic equation: x2 - y*x + 4 = 0
4. Now, solve for x using the quadratic formula, x = (-(-y) ± √((y2) - 4*1*4)) / (2*1)
5. Since we are looking for the inverse function, we only take the positive root which is x = (y + √(y2 - 16)) / 2

Therefore, the inverse function f-1(x) is given by f-1(x) = (x + √(x2 - 16)) / 2.

Answer 2

If f^(-1) is the inverse of f, then

f(f^(-1)(x)) = x

It looks like f(x) = (x + a)/x. This gives us

f(f^(-1)(x)) = (f^(-1)(x) + a)/f^(-1)(x) = x

Solve for f^(-1)(x) :

f^(-1)(x) + a = x f^(-1)(x)

f^(-1)(x) - x f^(-1)(x) = -a

(1 - x) f^(-1)(x) = -a

f^(-1)(x) = -a/(1 - x)

f^(-1)(x) = a/(x - 1)

If a = 4, then the inverse is f^(-1)(x) = 4/(x - 1).