Inverse Functions
Table of Contents
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 Functions make 3x + 2 back to x
, substitute y = 3x + 2,
then
So, the inverse of f(x) = 3x + 2 is f-1(y) =
Let’s put x = 3 on f(x):
f(3) = 3×3 + 2 = 11
Then put y = 11 on f-1(y):
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
| Inverse | Note |
| + <=> – | |
| x <=> ÷ | Can’t divide by zero |
| 1/x <=> 1/y | x and y not zero |
| x2 <=> √y | x and y ≥ 0 |
 | n 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 |
