The inverse function is a one-to-one function. Each object in the domain is map to each object in the range. To satisfy this requirement, the function f(x), should not have any turning point. That is, the gradient cannot be equal to zero. This can be verified by computing the the differentiated results - dy/dx) should not be equal to zero.Eg one-to-one function
Domain---------Range
1---------------3
2---------------5
3---------------7
4---------------9
Eg not one-to-one function
Domain----------Range
1
2---------------5
3---------------7
4---------------9
You can imagine the Range to be the mirror image of the Domain. The mirror is the line
y = x.
Inverse function of a linear equation
Eg, f(x) = x - 5
y = x - 5
exchange x to y and y to x,
x = y - 5
rearrange the equation,
x + 5 = y
y = x + 5
Inverse function of a quadratic equation
Eg, f(x) = x2 + 2x + 5
y = x2 + 2x + 5
y = (x + 1)2 + 4
exchange x to y and y to x
x = (y + 1)2 + 4
rearrange the equation
x - 4 = (y + 1)2
sq root(x - 4) = y + 1
[sq root(x - 4)] -1 = y
y = [sq root(x - 4)] - 1
Inverse function of a cubic equation
Eg f(x) = x3 + x - 1
y = x3 + x - 1
exchange x to y and y to x
x = y3 + y - 1
For cubic equation, it is impossible to re-arrange the equation to isolate y on the left hand side and the various x on the right hand side.f-1(9), 9 = y3 + y - 1
using trial and error method, determine the value of y.
if y = 1, 1 + 1 - 1 not equal to 9
if y = 2, 8 + 2 - 1 = 9
Therefore, f-1(9) = 2
Oops! sorry for the untidy graphs!
