Inverse Functions

Definition

The inverse is shown by putting a little “-1” after the function name, like this:

f-1(y), where y = f(x)

It says “f inverse of y

Here we have the function f(x) = 3x+2

The flow is

Inverse Function

Inverse Functions make 3x + 2 back to x

Inverse Function 2, substitute y = 3x + 2,

then

Inverse Function 3

So, the inverse of f(x) = 3x + 2 is f-1(y) = Inverse Function 4

Let’s put x = 3 on f(x):

f(3) = 3×3 + 2 = 11

Then put y = 11 on f-1(y):

Inverse Function 5

f-1(11) = = 3

So,

f-1(y)=f-1(f(x)) = x

Then, how about f (f-1(x)) ?

Because f-1(y)= x, then f (f-1(y)) = f (x) = y

f (f-1(y)) = y. Which means f (f-1(x)) = x

So

f-1(f(x)) = f (f-1(x)) = x

Inverses of Common Functions

So far is easy, because we know the inverse of Add is Subtract, and the inverse of Multiply is Divide. Ho about other functions?

List of inverse of Multiply is Divide

InverseNote
+ <=> – 
x <=> ÷Can’t divide by zero
1/x <=> 1/yx and y not zero
x2 <=> √yx and y ≥ 0
Inverse Function 6n not zero (different rules when n is even, odd, positive, or negative)
ex <=> ln(y)y > 0
ax <=> loga(y)y and a > 0
sin(x) <=> sin-1(y)-π/2 to +π/2
cos(x) <=> cos-1(y)0 to π
tan(x) <=> tan-1(y)-π/2 to +π/2

Learn More

Set-Builder Notation

Intervals

Injective, Surjective & Bijective

Function Transformations

Sets Index

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