MATH 1023/1043 Composite Functions 1. Given that g : x → 3x + 1 and f : y → 3y − 6, fill in the blanks below. (a) g(3) = . . . (b) f (10) = . . . (c) . . . (3) = 24 2. Fill in the blanks below. (a) The function that maps w to x is . . . (b) The function that maps x to y is . . . (c) The function that maps y to z is . . . (d) The function that maps w to y is . . . (e) The function that maps x to z is . . . (f) The function that maps w to z is . . . 1 3. The arrow diagrams show functions f and g. Determine the value of (a) f g(3) (b) gf (−5) 4. The diagram below shows two functions f and g. Determine (a) f (a) (c) g(r) (e) gf (b) (b) f (c) (d) gf (a) (f) gf (c) 2 5. The diagram shows two functions f : A → B and g : B → C and h = gf . Determine (c) h(2) × h(1) (a) f (2) + g(12) (b) h(3) − h(2) 6. Given that f (x) = x + 3 and g(x) = 4 − 8, find x (a) gf (1) (b) f g(1) (c) g 2 (3) 7. Given that f (x) = x+1 and g(x) = x + 5, find 2 (a) f g( 12 (b) gf ( 12 8. Given h(x) = 1 − x 9 and k(x) = , find 3 x (a) hk(x) (b) kh(x) 9. Given g(x) = 3x and f (x) = x + 4, find x−2 (a) gf (x) (b) f g(x) 3 (d) h(3) h(1) 10. Functions f and g are defined by f : x → 5 − 2x and g : x → 2x + 5. Find (a) f g(x) 1 (b) If f g(k) = 11k, show that k = − . 3 5 11. Given the functions f : x → x + 3, g : x → 4x − 5 and h : x → , find the composite x functions (a) gf h (b) f hg 2 12. Given the function f : x → 4x − , find the value of f 3 (1). x 4 13. Functions h and k are defined by h : x → and k : x → 3x − 4. Find the value of x x which has the same image under the function hk and kh. 14. A function f is defined by f (x) = 2x − m where m is a constant. Given that f 2 (2) = f (2), calculate the value of m. 3 and k : x → mx2 + n where m and n 15. Functions h and k are defined by h : x → x−3 are constants. (a) Given that k( 21 ) = 2 and kh(1) = 10, calculate the value of m and of n. (b) Obtain an expression for kh. 16. Functions h and k are defined by h : x → ax + b and k : x → x + 4 where a and b are constants. If hk(1) = 2 and kh(2) = 3, find (a) the values of a and b (b) kh( 12 ) 17. Given that f : x → mx + n where m > 0 and the function f 2 is f 2 : x → 4x + 12. Find (a) the function f 4 . (b) the values of m and n. (c) the value of x so that 6f (x) = f (2x − 3) 18. Given that g(x) = ax + b and the function g 3 (x) = 27x + 26. Find the values of a and b. 4 19. The diagram below shows the mapping of x to y by the function g(x) = 3x + a and mapping of y to z by the function f (y) = y − b, where a and b are constants. (a) Show that a − b = 2. (b) If f (−3) = g(−3), find the numerical value of a and of b. (c) Find the value of k that starts at 1 and ends at k. 20. The diagram below shows the mapping of x to y by the function f (x) = 3x + 5 and mapping of y to z by the function g(y) = 15 − 2y, where m and n are constants. Find (a) the value of m and of n. (b) the value of gf (4). 5 21. The diagram below shows the mapping of x to y by the function f (x) = px − 16 and q mapping of y to z by the function g(y) = where p and q are constants. y−5 Find (a) the value of p and of q. (b) the function that maps an element of x to an element of z. 22. The diagram below shows the function f that maps set A to set B and the function g that maps set B to set C. The functions f and g are defined as f : x → 5x and g(y) = y − 7 respectively. Determine (a) y in terms of x (b) z in terms of x (c) the composite function gf 23. (a) Given that f (x) = 2x + 5, show that f (x − 1) = 2x + 3 (b) Conversely, given that f (x − 1) = 2x + 3, show that f (x) = 2x + 5. 1−x 24. (a) Given that f (x) = 2x + 5, show that f = 6 − x. 2 1−x (b) Conversely, given that f = 6 − x, show that f (x) = 2x + 5. 2 25. Given f : x → x + 5 and f g : x → 8 − 3x, find the function g(x). 26. Given f : x → 6x − 2 and f g : x → 2x − 3, find the function g(x). 6 27. Given g(x) = 2 4 and gf (x) = , find the function f (x). x+1 x−6 28. Given f (x) = x + 2 and gf (x) = 2x + 3, find the function g(x). 29. Given f (x) = 2x − 7 and gf (x) = 6x − 8, find the function g(x). 30. A function f is defined by f : x → x−2 x+3 . Given that gf : x → , find the function x+1 2−x g(x). 31. The diagram below shows the mapping of function f followed by function g. Find (a) f (−2) (b) Determine the function g. 32. The diagram below shows the mapping of set A onto set B under the function f and the mapping of set B onto set C under the function g. Find (a) gf (4) (b) f g(−5) 7