# Properties of functions

## Real functions

### Real functions

Consider a function

$$y=f(x)$$

$$f(x)$$ is a real function if:

$$\forall x\in \mathbb{R} f(x) \in \mathbb{R}$$

### Support

$$f X\rightarrow R$$

Support of $$f$$ is $$x \in X$$ where $$f(x)\ne 0$$

### Monotonic functions

Calculus stationary points finding and monotonic functions

### Even and odd functions

#### Defining odd and even functions

An even function is one where:

$$f(x)=f(-x)$$

An odd function is one where:

$$f(x)=-f(-x)$$

#### Functions which are even and odd

If a function is even and odd:

$$f(x)=f(-x)=-f(-x)$$

$$f(x)=-f(x)$$

Then $$f(x)=0$$.

#### Scaling odd and even functions

Scaling an even function provides an even function.

$$h(x)=c.f(x)$$

$$h(-x)=c.f(-x)$$

$$h(-x)=c.f(x)$$

$$h(-x)=h(x)$$

Scaling an odd function provides an odd function.

$$h(x)=c.f(x)$$

$$-h(-x)=-c.f(-x)$$

$$-h(-x)=c.f(x)$$

$$-h(-x)=h(x)$$

#### Adding odd and even functions

Note than $$2$$ even functions added together makes an even function.

$$h(x)=f(x)+g(x)$$

$$h(x)=f(-x)+g(-x)$$

$$h(-x)=f(x)+g(x)$$

$$h(x)=h(-x)$$

And adding $$2$$ odd functions together makes an odd function.

$$h(x)=f(x)+g(x)$$

$$h(x)=-f(-x)-g(-x)$$

$$-h(-x)=f(x)+g(x)$$

$$-h(-x)=h(x)$$

#### Multiplying odd and even functions

Multiplying $$2$$ even functions together makes an even function.

$$h(x)=f(x)g(x)$$

$$h(-x)=f(-x)g(-x)$$

$$h(-x)=f(x)g(x)$$

$$h(-x)=h(x)$$

Multiplying $$2$$ odd functions together makes an even function.

$$h(x)=f(x)g(x)$$

$$h(-x)=f(-x)g(-x)$$

$$h(-x)=(-1).(-1.)f(x)g(x)$$

$$h(-x)=h(x)$$

### Concave and convex functions

#### Convex functions

A convex function is one where:

$$\forall x_1, x_2\in \mathbb{R} \forall t \in [0,1] [f(tx_1+(1-t)x_2 \le tf(x_1)+(1-t)f(x_2)]$$

That is, for any two points of a function, a line between the two points is above the curve.

A function is strictly convex if the line between two points is strictly above the curve:

$$\forall x_1, x_2\in \mathbb{R} \forall t \in (0,1) [f(tx_1+(1-t)x_2 < tf(x_1)+(1-t)f(x_2)]$$

An example is $$y=x^2$$.

#### Concave functions

A concave function is an upside down convex function. The line between two points is below the curve.

$$\forall x_1, x_2\in \mathbb{R} \forall t \in [0,1] [f(tx_1+(1-t)x_2 \ge tf(x_1)+(1-t)f(x_2)]$$

A function is strictly concave if the line between two points is strictly below the curve:

$$\forall x_1, x_2\in \mathbb{R} \forall t \in (0,1) [f(tx_1+(1-t)x_2 > tf(x_1)+(1-t)f(x_2)]$$

An example is $$y=-x^2$$.

#### Affine functions

If a function is both concave and convex, then the line between two points must be the function itself. This means the function is an affine function.

$$y=cx$$

## O

### Big $$O$$ and little-$$o$$ notation

#### Big $$O$$ notation

In big $$O$$ notation we are interested in t he size of a function as it getes larger. We ignore constant multiples.

$$cx\in O(x)$$

$$cx+b\in O(x)$$
$$x+x^2\in O(x^2)$$
$$f(x)\in O(g(x))$$