# Introduction to Twelve Basic Functions

## Key Questions

• The Sine function: $f \left(x\right) = \sin \left(x\right)$
The Cosine function: $f \left(x\right) = \cos \left(x\right)$
and
The Logistic function: $f \left(x\right) = \frac{1}{1 - {e}^{- x}}$
are the only function of the "Basic Twelve Functions" which are bounded above.

• Of the 12 Basic F|unctions only
The identity function: $f \left(x\right) = x$
and
The reciprocal function: $f \left(x\right) = \frac{1}{x}$
are their own inverses.

1. Identity: $f \left(x\right) = x$
2. Square: $f \left(x\right) = {x}^{2}$
3. Cube: $f \left(x\right) = {x}^{3}$
4. Reciprocal: $f \left(x\right) = \frac{1}{x} = {x}^{- 1}$
5. Square Root: $f \left(x\right) = \sqrt{x} = {x}^{\frac{1}{2}}$
6. Exponential: $f \left(x\right) = {e}^{x}$
7. Logarithmic: $f \left(x\right) = \ln \left(x\right)$
8. Logistic: $f \left(x\right) = \frac{1}{1 + {e}^{- x}}$
9. Sine: $f \left(x\right) = \sin \left(x\right)$
10. Cosine: $f \left(x\right) = \cos \left(x\right)$
11. Absolute Value: $f \left(x\right) = \left\mid x \right\mid$
12. Integer Step: $f \left(x\right) = \text{int} \left(x\right)$