Consider a function
\(y=f(x)\)
\(f(x)\) is a real function if:
\(\forall x\in \mathbb{R} f(x) \in \mathbb{R}\)
\(f X\rightarrow R\)
Support of \(f\) is \(x \in X\) where \(f(x)\ne 0\)
Calculus stationary points finding and monotonic functions
An even function is one where:
\(f(x)=f(-x)\)
An odd function is one where:
\(f(x)=-f(-x)\)
If a function is even and odd:
\(f(x)=f(-x)=-f(-x)\)
\(f(x)=-f(x)\)
Then \(f(x)=0\).
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.
\(h(x)=c.f(x)\)
\(-h(-x)=-c.f(-x)\)
\(-h(-x)=c.f(x)\)
\(-h(-x)=h(x)\)
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 \(2\) 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.
\(h(x)=f(x)g(x)\)
\(h(-x)=f(-x)g(-x)\)
\(h(-x)=(-1).(-1.)f(x)g(x)\)
\(h(-x)=h(x)\)
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 \(y=x^2\).
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 \(y=-x^2\).
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.
\(y=cx\)
In big \(O\) notation we are interested in t he size of a function as it getes larger. We ignore constant multiples.
\(cx\in O(x)\)
And addition of constants.
\(cx+b\in O(x)\)
If there are two terms and one is larger, we keep the largest.
\(x+x^2\in O(x^2)\)
More generally we write:
\(f(x)\in O(g(x))\)