find the inverse function of the following function

Question

f(x)= 4x/sqrrt(5x^2+4)

Answer 1

Follow the usual rule of interchanging x & y.

x = 4y / √(5y²+4)

squaring both sides and cross-multplying gives 5x²y² + 4x² = 16y²

and this leads to 4x² = y²(16-5x²).

Solving for y gives y = ±2x / √(16-5x²).

 

So this "inverse" is not a function (fails vertical line test). You may choose to drop the - option in front, to make the inverse exist.

 

As always, it's advisable to CHECK THIS ALGEBRA because of the possibility of mistakes (including typing).

Answer 2

 

 

y = (4x)/sqrt(5x^2 + 4)


x = (4y)/sqrt(5y^2 + 4)


x^2 = (16y^2)/(5y^2+4)

(5y^2 + 4)/(16y^2) = 1/x^2


5/16 + 1/(4y^2) = 1/x^2

1/(4y^2) = x^(-2) - 5/16

1/(2y) = x^(-2) - 5/16

1/y = 2x^(-2) - 5/8


1/y = 2/x^2 - 5/8

1/y = (2 - 5x^2)/(8x^2)

y = (8x^2)/(2-5x^2)