Question

f(x)=(x)/(2x-9) Find f⁻¹(x), where f⁻¹ is the inverse of f. Also state the domain and range of f⁻¹ in interval notation.

Answer 1

The inverse of f(x) = x / (2x - 9) is f⁻¹(x) = (9x) / (2x - 1). The domain of f⁻¹ is (-∞, 9/2) ∪ (9/2, +∞) and the range of f⁻¹ is (-∞, 0) ∪ (0, +∞) in interval notation.

Step-by-step explanation:

To find the inverse of the function f(x) = x / (2x - 9), we need to swap the roles of x and y and solve for y. First, we replace f(x) with y: y = x / (2x - 9). Next, we interchange x and y: x = y / (2y - 9). Now, we solve for y:

1. Cross multiply to eliminate the denominators: 2xy - 9x = y
2. Bring all y terms to one side: 2xy - y = 9x
3. Factor out y: y(2x - 1) = 9x
4. Divide both sides by (2x - 1): y = (9x) / (2x - 1)

So, the inverse of f(x) = x / (2x - 9) is: f⁻¹(x) = (9x) / (2x - 1).

The domain of f⁻¹ is the same as the range of f(x), which is all real numbers except x = 9/2. Therefore, the domain of f⁻¹ is (-∞, 9/2) ∪ (9/2, +∞) in interval notation.

The range of f⁻¹ is the same as the domain of f(x), which is all real numbers except y = 0. Therefore, the range of f⁻¹ is (-∞, 0) ∪ (0, +∞) in interval notation.