Find the inverse functions (f^-1(x)) of the functions below
f(x)=x^3+3
f(x)=(x+1)/(x-2) ; x≠2
To find the inverse function, we want to solve for x in terms of y.
1) f(x) = x^3 + 3
Let y = x^3 + 3
Swap x and y:
x = y^3 + 3
Now solve for y:
x - 3 = y^3
Take the cube root of both sides:
∛(x - 3) = y
So, the inverse function is f^(-1)(x) = ∛(x - 3).
2) f(x) = (x + 1)/(x - 2), x ≠ 2
Let y = (x + 1)/(x - 2)
Swap x and y:
x = (y + 1)/(y - 2)
Now solve for y:
xy - 2x = y + 1
xy - y = 2x + 1
Factor out y:
y(x - 1) = 2x + 1
Divide both sides by (x - 1):
y = (2x + 1)/(x - 1)
So, the inverse function is f^(-1)(x) = (2x + 1)/(x - 1), x ≠ 1.
To find the inverse function of f(x), we need to follow these steps:
For f(x) = x^3 + 3:
Step 1: Replace f(x) with y:
y = x^3 + 3
Step 2: Swap x and y:
x = y^3 + 3
Step 3: Solve for y:
x - 3 = y^3
y = (x - 3)^(1/3)
Therefore, the inverse function of f(x) = x^3 + 3 is:
f^(-1)(x) = (x - 3)^(1/3)
For f(x) = (x + 1)/(x - 2):
Step 1: Replace f(x) with y:
y = (x + 1)/(x - 2)
Step 2: Swap x and y:
x = (y + 1)/(y - 2)
Step 3: Solve for y:
x(y - 2) = y + 1
xy - 2x = y + 1
xy - y = 2x + 1
y(x - 1) = 2x + 1
y = (2x + 1)/(x - 1)
Therefore, the inverse function of f(x) = (x + 1)/(x - 2) is:
f^(-1)(x) = (2x + 1)/(x - 1)