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# Compositions of Functions

Consider functions, f: A â†’ B and g: B â†’ C. The composition of f with g is a function from A into C defined by (gof) (x) = g [f(x)] and is defined by gof.

Â Â Â To find the composition of f and g, first find the image of x under f and then find the image of f (x) under g.

**Example1:**

Consider the function f = {(1, a), (2, a), (3, b)} and g = {(a, 5), (b, 7)} as in figure. Find the composition of gof.

**Solution:** The composition function gof is shown in fig:

(gof) (1) = g [f (1)] = g (a) = 5, (gof) (2) = g [f (2)] = g (a) = 5 (gof) (3) = g [f (3)] = g (b) = 7.

**Example2:** Consider f, g and h, all functions on the integers, by f (n) =n^{2}, g (n) = n + 1 and h (n) = n â€“ 1.

Determine **(i)** hofog Â Â Â **(ii)** gofoh Â Â Â **(iii)** fogoh.

**Solution: **

(i) hofog (n) = n + 1, hofog (n + 1) = (n+1)^{2}h [(n+1)^{2}] = (n+1)^{2}- 1 = n^{2}+ 1 + 2n - 1 = n^{2}+ 2n. (ii) gofoh (n) = n - 1, gof (n - 1) = (n-1)^{2}g [(n-1)^{2}] = (n-1)^{2}+ 1 = n^{2}+ 1 - 2n + 1 = n^{2}- 2n + 2. (iii) fogoh (n) = n - 1 fog (n - 1) = (n - 1) + 1 f (n) = n^{2}.

**Note:**

- If f and g are one-to-one, then the function (gof) (gof) is also one-to-one.
- If f and g are onto then the function (gof) (gof) is also onto.
- Composition consistently holds associative property but does not hold commutative property.

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