Final answer:
To find f⁻¹(2), we derived the inverse function from f(x) = (5x-2)/(4-5x), setting y=f(x) and solving for x in terms of y. After swapping x and y, solving for y, and simplifying, we plugged in x=2 into the inverse to find that f⁻¹(2) = 2/3.
Step-by-step explanation:
To find f⁻¹(2), we first need to express the function f(x) in a way that makes x the subject. The function given is f(x) = (5x-2)/(4-5x). To find its inverse, we set f(x) equal to y and solve for x:
y = (5x-2)/(4-5x)
Now we swap x and y:
x = (5y-2)/(4-5y)
Then, by solving for y, we find the inverse function:
- Multiply both sides by (4-5y) to get rid of the fraction:
- x(4-5y) = 5y-2
- Expand the left side and gather like terms:
- 4x - 5xy = 5y - 2
- Add 5xy to both sides and add 2 to both sides:
- 4x + 2 = 5y + 5xy
- Factor out y on the right side:
- 4x + 2 = y(5+5x)
- Finally, divide both sides by (5+5x) to isolate y:
- y = (4x + 2)/(5 + 5x)
Now that we have the inverse function, we substitute x=2 into the inverse to find f⁻¹(2):
f⁻¹(2) = (4(2) + 2)/(5 + 5(2))
f⁻¹(2) = (8 + 2)/(5 + 10)
f⁻¹(2) = 10/15
f⁻¹(2) = 2/3