The transcript details a mathematics lecture on functions and sets, specifically discussing the concept of composite functions. The speaker explains how to compose functions using diagrams representing different sets (A, B, C) and applies functions \( f \), \( g \), and \( h \) to elements within these sets. The lecture includes examples of calculating images of elements under these functions and emphasizes the order of composition. Additionally, numerical examples are provided to illustrate the process of function composition. The key takeaway is understanding how composite functions work and how to apply them in various mathematical contexts.
Here are the key facts from the text:
1. There are sets A, B, and C, and functions f and g that map elements between these sets.
2. Function f maps elements from set A to set B.
3. Function g maps elements from set B to set C.
4. The composite function h is defined as h(x) = g(f(x)).
5. The domain of function h is set A, and its range is set C.
6. The composite function h can be written as h(x) = g(f(x)) or h(x) = g(f(x)) = g(2x + 3).
7. When applying the composite function h to an element x, we first apply function f to x, then apply function g to the result.
8. The function f is defined as f(x) = x + 3.
9. The function g is defined as g(x) = 2x.
10. The composite function h is defined as h(x) = g(f(x)) = 2(x + 3).
11. The value of h(2) is 11.
12. The composite function of g and f is not the same as the composite function of f and g.
13. The function gdf(x) is defined as g(f(x)).
14. The function f(x) is defined as x squared - 1.
15. The function g(x) is defined as 3x + 2.
16. The composite function gdf(x) is defined as g(f(x)) = 3(x squared - 1) + 2.
17. The value of gdf(2) is 8.
18. The function gdf(x) can also be written as 3x squared - 3 + 2.
19. The value of gdf(2) is 11.
20. The function egd(x) is defined as e(g(d(x))).
21. The value of egd(2) is 66/23.