1 00:00:00,557 –> 00:00:07,613 The mathematical symbol of today is this small circle, we use for the composition of maps or functions.

2 00:00:08,371 –> 00:00:15,378 It’s a binary operation, where we have one map “f” on the right hand side and another map “g” on the left hand side.

3 00:00:15,886 –> 00:00:21,648 Indeed the composition is very basic, but it’s an important notion throughout mathematics.

4 00:00:22,271 –> 00:00:25,275 It simply merges two maps into a new one.

5 00:00:25,943 –> 00:00:30,640 and the definition can simply be given when we put in an element “x”.

6 00:00:31,443 –> 00:00:35,829 First f is applied to x and then g is applied on this output.

7 00:00:36,686 –> 00:00:39,908 In other words we put f(x) into g.

8 00:00:40,400 –> 00:00:42,914 So we have g(f(x)).

9 00:00:43,929 –> 00:00:50,886 Now to remember this order, it’s always good to have a visualization of the sets, where the maps act on, in mind.

10 00:00:51,829 –> 00:00:55,382 First we have f that maps from left to the middle.

11 00:00:56,014 –> 00:00:59,813 and then we have g that maps from the middle set to the right set.

12 00:01:00,800 –> 00:01:06,486 Therefore the new map g composed with f goes immediately from left to right.

13 00:01:07,014 –> 00:01:09,929 Ok and that explains the composition symbol.

14 00:01:10,531 –> 00:01:14,844 If you want to know more i have a whole video about the composition of maps.

15 00:01:15,229 –> 00:01:18,084 and with this i hope i see you in the next video.

16 00:01:18,284 –> 00:01:19,273 Bye!

Q1: Let $f,g: \mathbb{R} \rightarrow \mathbb{R}$. How is $f \circ g$ defined?

A1: $(f \circ g) (x) = f(g(x))$ for all $x \in \mathbb{R}$.

A2: $(f \circ g) (x) = f(x) \cdot g(x)$ for all $x \in \mathbb{R}$.

A3: $(f \circ g) (x) = f(x) + g(x)$ for all $x \in \mathbb{R}$.

A4: $(f \circ g) (x) = f(x) - g(x)$ for all $x \in \mathbb{R}$.

Q2: Let $f,g: \mathbb{R} \rightarrow \mathbb{R}$ be given by $g(x) = x^2$ and $f(x) = \sin(x)$. What is $f \circ g$ in case?

A1: $x \mapsto \sin(x^2)$

A2: $x \mapsto \sin^2(x)$

A3: $x \mapsto (\sin(x))^2$

A4: $x \mapsto \sin(x^4)$