To find inverse of a linear function, we follow the steps given below.
Step 1 :
The given equation will be in the form y =, Derive the equation for x = .
Step 2 :
After solving for x, change x as f-1(x) and y as x.
Relationship between f(x) and and f-1(x) :
Domain of the function f(x) = Range of f-1(x)
Range of the function f(x) = Domain of f-1(x)
Find the inverse of the given function :
Problem 1 :
f(x) = (5/6)x - 1/3
Solution :
f(x) = (5/6)x - 1/3
Let y = f(x)
Solving for x :
y = (5/6)x - (1/3)
y + (1/3) = (5/6)x
(3y + 1)/3 = (5/6)x
(6/3)(3y + 1) = 5x
2(3y + 1) = 5x
x = (2/5) (3y + 1)
Replacing x as f-1(x) and x as y :
f-1(x) = (2/5) (3x + 1)
Problem 2 :
f(x) = 2x/3
Solution :
f(x) = 2x/3
Let f(x) = y
y = 2x/3
Solving for x :
x = 3y/2
Replacing x as f-1(x) and x as y :
f-1(x) = 3x/2
Problem 3 :
f(x) = -5x - 5/4
Solution :
f(x) = -5x - 5/4
Let f(x) = y
y = -5x - 5/4
Solving for x :
y + 5/4 = -5x
x = (-1/5)(4y + 5)/4
x = (-1/20)(4y + 5)
x = (-y/5) + (1/4)
x = (-y/5) + (1/4)
Replacing x as f-1(x) and x as y :
f-1(x) = (-x/5) + (1/4)
Problem 4 :
g(x) = 6x - 5
Solution :
g(x) = 6x - 5
Let g(x) = y
y = 6x - 5
Solving for x :
6x = y + 5
x = (1/6)(y + 5)
Replacing x as g-1(x) and x as y :
So, the inverse function is,
g-1(x) = (1/6)x + (5/6)
Problem 5 :
g(x) = 4 + x/2
Solution :
g(x) = 4 + x/2
Let g(x) = y
y = 4 + x/2
Solving for x :
y - 4 = x/2
x = 2(y - 4)
x = 2y - 8
Replacing x as g-1(x) and x as y :
g-1(x) = 2x - 8
So, the required inverse function is
g-1(x) = 2x - 8.
Problem 6 :
g(x) = 2 + 1/3x
Solution :
g(x) = 2 + 1/3x
Let g(x) = y
y = 2 + 1/3x
Solving for x :
y - 2 = (1/3) x
3(y - 2) = x
x = 3y - 6
Replacing x as g-1(x) and x as y :
g-1(x) = 3x - 6
So, the required inverse function is,
g-1(x) = 3x - 6
Problem 7 :
h(x) = 4x + 3
Solution :
h(x) = 4x + 3
Let h(x) = y
y = 4x + 3
Solving for x :
y - 3 = 4x
x = (1/4)(y - 3)
Replacing x as g-1(x) and x as y :
h-1(x) = (1/4) (x - 3)
So, the required inverse function is
h-1(x) = (x - 3)/4.
Problem 8 :
h(x) = 2x + 2
Solution :
h(x) = 2x + 2
Let h(x) = y
y = 2x + 2
Solving for x :
2x = y - 2
x = (1/2) (y - 2)
Replacing x as h-1(x) and x as y :
h-1(x) = (1/2) (x - 2)
So, the required inverse function is,
h-1(x) = (1/2) (x - 2)
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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