Question

The inverse of f(x) = (2x-1)(x+5) may be written in the form f⁻¹(x) = (ax+b)(cx+d) , where a , b , c , and d are real numbers. Find (a)(c) .

Answer 1

To find the inverse of the function f(x) = (2x-1)(x+5), we interchange x and y, solve for y, and rewrite the equation in the form ax² + bx + c = 0. The value of (a)(c) is -2x - 10.

Step-by-step explanation:

To find the inverse of the function f(x) = (2x-1)(x+5), we need to interchange x and y and then solve for y.
Step 1: Replace f(x) with y:
y = (2x-1)(x+5)
Step 2: Interchange x and y:
x = (2y-1)(y+5)
Step 3: Solve for y:
x = 2y² +9y - y - 5
x = 2y² +8y - 5
Step 4: Rearrange the equation:
2y² + 8y - (x+5) = 0

This equation can be written in the form ax² + bx + c = 0, where a = 2, b = 8, and c = -(x+5).
Using this form, we can find ac by multiplying the values of a and c:
(a)(c) = (2)(-(x+5)) = -2x - 10

Therefore, (a)(c) = -2x - 10.