1. Composite functions When two or more functions are combined, so that the output from the first function becomes the input to the second function, the result is called a composite function or a function of a function. Consider f(x) = 2x -1 with the domain {1, 2, 3, 4} and g = x 2 with domain the range of f. 12341234 f(x) = 2x - 1g(x) = x 2 13571357 1 9 25 49 Domain of f Range of f Domain of g Range of g gf(x)

2. fg and gf In general, the composite function fg and gf are different functions f(x) = 2x – 1 and g(x) = x 2 gf(x) 1 st function applied 2 nd function applied gf(x) = (2x – 1) 2 e.g. gf(3) = 25 fg(x) = 2x 2 - 1 e.g. fg(3) = 17

3. Examples Find f(3) and f(-1) f(3) = (4  3 – 1) 2 = 121f(-1) = (4  -1- 1 )2=(- 5) 2 = 25 Find (i) gf(2) (ii) gg(2) (iii) fg(2) gff(2) (i) gf(x) = 2x 2 – 1  gf(2) = 2  2 2 – 1 = 7 (ii) gg(x) = 2(2x – 1)– 1  gg(2) = 2(2  2-1) – 1 = 5 (iii) fg(x) = (2x – 1) 2  fg(2) = (2  2 – 1) 2 = 9 (iv) gff(x) = 2x 4 - 1  gff(2) = 2  2 4 – 1 = 31

4. Examples Break the following functions down into two or more components. (i) f(x) = 2x + 3 and g(x) = x 2  fg(x) = 2x 2 + 3 (ii) f(x) =  x, g(x) = x - 3 and h = x 4  hgf(x) = (  x – 3) 4 Find the domain and corresponding range of each of the following functions. (i) Domain: x  2 range f(x)  2 (ii) Domain: x  0 range f(x)  0

5. Examples (i) x  x 2 + 4(ii) x  x 6 (iii) x  3x + 12 (iv) x  9x 2 + 4 (v) x  (3x + 4) 2 (vi) 3x + 12 Express the following functions in terms of f, g and h as appropriate. (i) fh(x) = x 2 + 4(ii) hhh(x) = x 6 (iii) gf(x) = 3x + 12(iv) fggh(x) = 9x 2 + 4 (v) hgf(x) = (3x + 4) 2 (vi) fffg(x) = 3x + 12

6. Inverse functions The inverse function of f maps from the range of f back to the domain. f has the effect of ‘double and subtract one’ the inverse function (f -1 ) would be ‘add one and halve’. f(x) f -1 (x) range of f domain of f -1 domain of f range of f -1 AB The inverse function f -1 only exists if f is one – one for the given domain.

7. Graph of inverse functions f(x) = 2x - 1 y = x f(2) = 3  (2, 3) f -1 (3) = 2  (3, 2) y x In general, if (a, b) lies on y = f(x) then (b, a) on y = f – 1 (x). For a function and its inverse, the roles of x and y are interchanged, so the two graphs are reflections of each other in the line y = x provided the scales on the axes are the same.

8. Finding the inverse function f -1 Put the function equal to y. Rearrange to give x in terms of y. Rewrite as f – 1 (x) replacing y by x. Example find the inverse f - 1 (x).

9. Examples find the inverse f - 1 (x). x 2 -2x = y (x – 1) 2 – 1 = y (x – 1) 2 = y + 1 x – 1 =  (y + 1) x =  (y + 1)+ 1 f -1 (x) =  (x + 1)+ 1 x  - 1

10. Examples find the inverse f - 1 (x).