Question

The one-to-one function g is defined bel g(x)=(8x-1)/(5x+9) Find g^(-1)(x), where g^(-1) is the inverse of Also state the domain and range of g^(-1)

Answer 1

To find the inverse function g^(-1)(x) of the given one-to-one function g(x), swap the roles of x and y, solve for y, and rearrange the equation. The domain of g^(-1) is the same as the range of g(x), and the range of g^(-1) is the same as the domain of g(x).

Step-by-step explanation:

To find the inverse function g^(-1)(x) of the given one-to-one function g(x), we can swap the roles of x and y and solve for y:

x = (8y - 1) / (5y + 9)

Next, we can multiply both sides by (5y + 9) to eliminate the denominator:

x(5y + 9) = 8y - 1

Expanding and rearranging the equation:

5xy + 9x = 8y - 1

Bringing all the y terms to one side:

5xy - 8y = -9x - 1

Factoring out y:

y(5x - 8) = -9x - 1

Dividing both sides by (5x - 8) to solve for y:

y = (-9x - 1) / (5x - 8)

This is the inverse function g^(-1)(x). The domain of g^(-1) is the same as the range of the original function g(x), and the range of g^(-1) is the same as the domain of g(x).