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