Find the inverse f^-1(x) of the following functions
f(x)=3x-7
f(x)=x^2/(x^2-2)
To find the inverse of a function, we need to switch the x and y variables and solve for y.
For f(x) = 3x - 7:
Let y = 3x - 7
Swap x and y: x = 3y - 7
Solve for y: x + 7 = 3y
Divide both sides by 3: (x + 7) / 3 = y
The inverse function is f^-1(x) = (x + 7) / 3.
For f(x) = x^2 / (x^2 - 2):
Let y = x^2 / (x^2 - 2)
Swap x and y: x = y^2 / (y^2 - 2)
Multiply both sides by (y^2 - 2): x(y^2 - 2) = y^2
Expand: xy^2 - 2x = y^2
Rearrange terms: y^2 - xy^2 = 2x
Factor out y^2: y^2(1 - x) = 2x
Divide both sides by (1 - x): y^2 = 2x / (1 - x)
Take the square root of both sides: y = ±√[2x / (1 - x)]
The inverse function has two branches: f^-1(x) = ±√[2x / (1 - x)].
To find the inverse of a function, we need to swap the roles of x and y and solve for y.
1. For the function f(x) = 3x - 7:
Step 1: Replace f(x) with y: y = 3x - 7.
Step 2: Swap x and y: x = 3y - 7.
Step 3: Solve for y:
x + 7 = 3y
(x + 7)/3 = y
So, the inverse function is f^(-1)(x) = (x + 7)/3.
2. For the function f(x) = x^2/(x^2 - 2):
Step 1: Replace f(x) with y: y = x^2/(x^2 - 2).
Step 2: Swap x and y: x = y^2/(y^2 - 2).
Step 3: Solve for y:
x(y^2 - 2) = y^2
xy^2 - 2x = y^2
xy^2 - y^2 = 2x
y^2(xy - 1) = 2x
y^2 = 2x / (xy - 1)
y = sqrt(2x / (xy - 1)) or (-sqrt(2x / (xy - 1)))
So, the inverse function is f^(-1)(x) = sqrt(2x / (xy - 1)) or (-sqrt(2x / (xy - 1))).