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Composite functions

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.

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Composite functions

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  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 = x2 with domain the range of f. gf(x) f(x) = 2x - 1 g(x) = x2 1 9 25 49 1 2 3 4 1 3 5 7 Range of f Domain of g Domain of f Range of g

  2. fg and gf In general, the composite function fg and gf are different functions f(x) = 2x – 1 and g(x) = x2 gf(x) 2nd function applied 1st function applied gf(x) = (2x – 1)2 e.g. gf(3) = 25 e.g. fg(3) = 17 fg(x) = 2x2 - 1

  3. Examples Find f(3) and f(-1) f(3) = (43 – 1)2 = 121 f(-1) = (4-1- 1)2=(-5)2= 25 Find (i) gf(2) (ii) gg(2) (iii) fg(2) gff(2) (i) gf(x) = 2x2 – 1  gf(2) = 222 – 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) = 2x4 - 1 gff(2) = 224 – 1 = 31

  4. Examples Break the following functions down into two or more components. (i) f(x) = 2x + 3 and g(x) = x2 fg(x) = 2x2 + 3 (ii) f(x) = x , g(x) = x - 3 and h = x4  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 Express the following functions in terms of f, g and h as appropriate. • x  x2 + 4 (ii) x  x6 (iii) x  3x + 12 • (iv) x  9x2 + 4 (v) x  (3x + 4)2 (vi) 3x + 12 (i) fh(x) = x2 + 4 (ii) hhh(x) = x6 (iii) gf(x) = 3x + 12 (iv) fggh(x) = 9x2 + 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) range of f domain of f -1 domain of f range of f -1 A B f -1(x) The inverse function f -1 only exists if f is one – one for the given domain.

  7. y x Graph of inverse functions f(x) = 2x - 1 f(2) = 3  (2, 3) y = x f -1(3) = 2  (3, 2) 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). x2 -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).

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