Find the inverse of each of these functions
a)f(x)=x/x+2
b)f(x)=2x/5-x
c)f(x)=3x/2x+1
x =2y/5 -y
5x = 2y -5y
5x = -3y
F(x) = -5x/3
1st one:
if f(x) = x/(x+2) , then y = x/(x+2)
step1: interchange the x and y variables to get the inverse as
x = y/(y+2)
step 2: solve this new equation for y
x = y/(y+2)
xy + 2x = y
xy - y = -2x
y(x-1) = -2x
y = -2x/(x-1) or y = 2x/(1-x)
f^-1 (x) = 2x/(1-x)
now do the same for others following these two steps.
To find the inverse of a function, we need to switch the roles of x and y and solve for the new y.
a) f(x) = x/(x + 2)
Step 1: Replace f(x) with y.
y = x/(x + 2)
Step 2: Swap x and y.
x = y/(y + 2)
Step 3: Solve for y.
x(y + 2) = y
xy + 2x = y
xy - y = -2x
y(x - 1) = -2x
y = -2x / (x - 1)
So, the inverse of f(x) = x/(x + 2) is f^(-1)(x) = -2x / (x - 1).
b) f(x) = (2x)/(5 - x)
Step 1: Replace f(x) with y.
y = (2x)/(5 - x)
Step 2: Swap x and y.
x = (2y)/(5 - y)
Step 3: Solve for y.
x(5 - y) = 2y
5x - xy = 2y
5x = 2y + xy
5x = y(2 + x)
y = (5x) / (2 + x)
So, the inverse of f(x) = (2x)/(5 - x) is f^(-1)(x) = (5x) / (2 + x).
c) f(x) = (3x)/(2x + 1)
Step 1: Replace f(x) with y.
y = (3x)/(2x + 1)
Step 2: Swap x and y.
x = (3y) / (2y + 1)
Step 3: Solve for y.
x(2y + 1) = 3y
2xy + x = 3y
2xy - 3y = -x
y(2x - 3) = -x
y = -x / (2x - 3)
So, the inverse of f(x) = (3x)/(2x + 1) is f^(-1)(x) = -x / (2x - 3).