Download Composite Functions – Problems

Composite Functions – Problems
1. Let f  1,   ,  ,3 ,  0,4  , *,7 
and g   3,2  ,  7,* ,  5,1
with k ( x)  f ( g ( x))
(a) Find D k
(b) Find R g
Df
(c) Find R k
(d) Draw a set diagram with all mappings.
2. Let f ( x)  x 3 and g ( x)  x 2  x  1
(a) Express h( x)  g ( f ( x)) in terms of x.
(b) D h  ?
(c) R h  ?
3. Let f ( x)  x 4  x 2 and g ( x)  x 2  x  1
(a) f ( g ( x))  ?
(b) D f g  ?
4. Let m( x)  log 2 x, n( x)  x 2  1, h( x)  m( n( x)).
(a) D n  ? (b) D h  ? (c) R n
D m  ? (d) R m  ? (e) R h  ?
5. m( x)  cos x,0  x  2 , n( x)  2 x  1, h  n m, p  m n
(a) R m
D n  ? (b) D h  ? (c) R h  ? (d) D p  ? (e) R p  ?
6. Let m( x)  3x and n( x)  1
and h( x)  n(m( x)).
x 1
(a) Dm  ? (b) R m Dn  ? (c) Dh  ? (d) R n  ? (e) R h  ?
1
7. Consider the graph of n(x) =
. (Compare with y = 1/x)
x 1
What is the range of values for n(x) if x > 0 and x  1 ?
(a) Show graphically (b) Show analytically (considering (i) x > 1 and (ii) 0 < x < 1)
8. f ( x)  1  x , g ( x)  x  1,find D f g .
9. Let f ( x)  Sin 1 x, g ( x)  sin x, m( x)  g ( f ( x)).
Find: (a) m( x)  ? (b)Df  ? (c)Dm  ? (d) R f
Dg  ?
(e) R g  ? (f) R m  ? (g) m(0)  ? (h) m( 3)  ? (i) m( 4)  ?
10. Let f ( x)  Cos 1 x, g ( x)  Sin x, h( x)  g ( f ( x)).
Find: (a) h( x)  ? (b)Df  ? (c)Dh  ? (d) R f Dg  ?


(e) R g  ? (f) R h  ? (g) h(0)  ? (h) h 1 2   ? (i) h  3 2  ?
1
1
11. Let f ( x)  1  , g ( x)  1  , find (and give the domain restrictions for functions):
x
x
(a) f ( g ( x))  ? (b)g ( f ( x))  ? (c) f ( g (1  x))  ? (d) g ( f (1  1x ))  ?
(e) R g(f(x))  ? (f) R f(g(x))  ?
Composite Functions – Problems (continued)
12. Let f ( x)  x 2  36 and g ( x)  4 x , find the domain of g f .
13. Let u( x)  x and  ( x)  121  x 2 , find Du ( ( x )) .
14. Express the following functions as composites of two functions.
For example/ Do you see the following function as a composite of…?
y  tan( x) = ?
Solution: Let g ( x)  x and f ( x)  tan  x  , then y  g ( f ( x))
(a) y  tan  x
(b) y   log 2 x 
15. Express the following function as a composite of three functions (triple composite).
y  tan( x) = ?
16. Of the two translations given below, which is or may be considered a composite function
more than a sum or difference function?
f(x)  sin(x  6 )
F(x) = sin(x) + d
vs
17. Of the two ‘change of scales’ give below, which is or may be considered a composite
function more than a product or quotient function?
F(x) = 3sin x
vs
f(x) = sin(3x)
x 1
18. Let g ( x)  2  x and f ( x)  2 , find D f g and R f g and express f g in terms of x.
3
1
and g ( x)  ln( x  1), with h  g f . Find Dh .
x 1
20. Let f ( x)  x 2  1 and g ( x)   x , with h( x)  g ( f ( x)). Find D h and R h .
19. Let f ( x) 
Composite Functions – Quiz 1
1. s(x) = 2x and t(x) = 3x
2
(a) s(t(x)) = _________ (b) Dt = ________ (c) Rs(t(x)) = ________
2. g ( x)  2 x  2
h( x )  2 x  3
(a) g(h(x)) = _________ (b) Dg(h(x)) = _______ (c) Rg(h(x)) = ________
3. g ( x)  x
f ( x)  1/ x
(a) f(g(x)) = _________ (b) Df(g(x)) = ________ (c) Rf(g(x)) = ________
4. Given 0  x  2 , for what values of x is the following function defined?
y  cos  log 2 x 
Composite Functions – Solutions
1.
Where k  f g ,
(a) Dk  7,5 (b) R g
D f  *,1 (c) R k  ,7
Can you imagine the Dk and Rk circled in the above diagram?
2. (a) h(x) = x6 + x3 – 1
(b) Dh = Reals
(c) With a graphing calculator and/or a bit of calculus, we could determine that…
5

Rh =  y y    (Don’t worry, we can’t be expected to find this. Yet!)
4

3. (a) f(g(x)) = (x2 + x – 1)4 – (x2 + x – 1)2
What if you had to expand this? Ugh!
What if you were asked to find the zeroes?!?
Hmm…
Factor out (dividing out would be poor technique!)…
2
(x + x – 1)2((x2 + x – 1)2 – 1) = 0 (Do you see the ‘difference of 2 squares?)
(x2 + x – 1)2((x2 + x – 1) + 1)((x2 + x – 1) – 1) = 0
(x2 + x – 1)2(x2 + x)(x2 + x – 2) = 0 (Isn’t all this factoring and algebra fun?)
(x2 + x – 1)2x(x + 1)(x – 1)(x + 2) = 0 (What about that first factor?)
Quadratic formula… b2  4ac  12  4(1)(1)  5  0 darn…
1  5
2
Zeroes? x = 0, -1, 1, -2, 12
x
5
(6 distinct roots from an 8th degree equation!)
(b) Df g 
4. (a)
(b) {x | x < -1 or x > 1} (c) {z | z > 0} (d)
(e)
2
5. (a) {1} (b) {0, 2 } (c) {0} (d)  x 1  x    1 (e)  y y  1
6. (a)
(b) z z  0, z  1 (c)  x x  0 (d) z z  0 (e)  y y  1  y  0
Composite Functions – Solutions (continued)
1
1
7. Notice that n(x) 
is the hyperbola, y  , shifted one unit to the right.
x 1
x
(a) Oops! I forgot the x-restrictions, x >0 especially! The 2nd graph is better!
Still working on ‘open dots’!
(b) Remember, we’re trying to show that the range is (i) y > 0 or (ii) y < -1.
Case (i) x > 1 means x – 1 > 0, hence n(x) = 1/(x-1) > 0 (y > 0).
Case (ii) 0 < x < 1 means x > 0 and x < 1. (We want to get 1/(x-1) < -1, so…don’t ask!)
x > 0  x-1 > -1 We’re almost there. Just flip ‘x-1’ and…
and x < 1  x-1 < 0 but ‘x-1’ is negative. What happens to the ‘>’ sign above?
(Hmm… There is an inequality theorem which says: a  b  1a  b1
but it requires: a, b >0. Okay here’s the trick: Multiply by ‘-1’ to
make each side of ‘x-1 > -1’ positive and then use the theorem!)
-(x-1) < -(-1)
Notice the switch in the inequality sign.
-1
Using the theorem: a  b  1a  b1 , -(x-1) is positive.
1
x 1
1
n(x) =
Multiplying by ‘-1’ we finally get: y < -1.
 1
x 1
8. x 1  x  2
9. (a) m(x) = x, x  1 (b)  x x  1 (c)  x x  1 (d)  y y  2 
(e)  y y  1 (f)  y y  1 (g) 0 (h) not defined (i)  4
10. (a) 1  x 2 ,0  x  1 (b)  x x  1 (c) x 0  x  1 (d)  y 0  y   2
(e)  y y  1 (f) x 0  x  1 (g) 1 (h)
3
(i) undefined
2
2x  1
1
2x  1
x
11. (a)
, x  0,1 (b)
, x  0, 1 (c)
, x  0,1 (d)
, x  0,1, 12
x 1
x 1
x
2x  1
(e) R g f   y y  0,1 (f) R f g   y y  1,2
Composite Functions – Solutions (continued)
12. (, 6] [6, )
13. [-11,11]
14. (a) Let g(x) = tan x and f(x) =  x , then y = g(f(x))
(b) Let f(x) = log2 x and g(x) = x3, then y = g(f(x))
15. (a) Let f(x) =  x (a linear function), g(x) = x (square root function), and
h(x) = tan x (tangent function), then y  tan( x) = g(h(f(x))).
16. f(x)  sin(x  6 ) is a composition of y = x - 6 (a difference function) and
y = sin x (sine function).
17. f(x) = sin(3x) is a composition of y = 3x (tripling function) and y = sin x (sine)
18. D  x x  2,R   y y  12
19. Dh = {x | x > 1 or x < 0}
20. Dh = x 1  x  1,R   y 0  y  1
Composite Functions – Quiz 1 - Answers
1. s(x) = 2x2 and t(x) = 3x
(a) s(t(x)) =
18x2
2. g ( x)  2 x  2
(b) Dt = Reals
(c) Rs(t(x)) =  y y  0
h( x )  2 x  3
(a) g(h(x)) =  y y  0 (b) Dg(h(x)) =  x x  1 (c) Rg(h(x)) =  y y  0
3. g ( x)  x
f ( x)  1/ x
(a) f(g(x)) =
x
1
or
x
x
(b) Df(g(x)) = x x  0 (c) Rf(g(x)) =  y y  0
4. Given 0  x  2 , for what values of x is the following function defined?
y  cos  log 2 x 
x 0  x  2 