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Composite and Inverse Functions Worksheet

Composite and Inverse Functions Worksheet
  • Page 1
 1.  
Given f(x) = 7 + 5x - 2x2 and g(x) = 2x + 3, evaluate (f o g)(x).
a.
- 4x2 + 10x + 11
b.
- 8x2 - 14x + 4
c.
- 4x3 + 4x2 + 29x + 21
d.
- 4x2 + 10x + 17
e.
8x2 + 34x + 40


Solution:

(f o g)(x) = f[g(x)]
[Composition of f(x) and g(x).]

= f [2x + 3]
[Replace g(x) = 2x + 3.]

= 7 + 5(2x + 3) - 2(2x + 3)2
[Replace x with (2x + 3) in f(x).]

= 7 + 10x + 15 - 2(4x2 + 12x + 9)

= 7 + 10x + 15 - 8x2 - 24x - 18

= - 8x2 - 14x + 4
[Simplify.]


Correct answer : (2)
 2.  
Given f(x) = 5x2 - 6x and g(x) = 8x - 1, evaluate (f o g)(x).
a.
320x2 - 128x + 11
b.
320x2 - 80x - 1
c.
320x2 - 32x - 1
d.
40x2 - 48x - 1
e.
40x3 - 53x2 + 6x


Solution:

(f o g)(x) = f[g(x)]
[Composition of f(x) and g(x).]

= f [8x - 1]
[Replace g(x) = 8x - 1.]

= 5(8x - 1)2 - 6(8x - 1)
[Replace x with (8x - 1) in f(x).]

= 5(64x2 - 16x + 1) - 48x + 6

= 320x2 - 80x + 5 - 48x + 6

= 320x2 - 128x + 11
[Simplify.]


Correct answer : (1)
 3.  
Given f (x) = x+1, evaluate (f o f)(8).
a.
6
b.
7-1
c.
3
d.
9
e.
2


Solution:

( f o f )(x) = f [f (x)]
[Composition of f (x) and f (x).]

( f o f )(8) = f [f (8)]
[Replace x = 8.]

= f (8+1)

= f (9) = f (3)

= 3+1 = 4 = 2
[Replace x = 3 in f (x).]


Correct answer : (5)
 4.  
Given h(x) = x2 - 9, evaluate (h o h)(13).
a.
493
b.
16
c.
160
d.
7
e.
169


Solution:

(h o h)(x) = h[h(x)]
[Composition of h(x) and h(x).]

(h o h)(13) = h[h(13)]

= h[(13)2 - 9]
[Replace x with 13 in h(x).]

= h[13 - 9] = h(4)
[Simplify.]

= (4)2 - 9
[Replace x with 4 in h(x).]

= 7
[Simplify.]


Correct answer : (4)
 5.  
Given g(x) = 4x - 3 and h(x) = x5 + 2 5, evaluate (g o h)(- 3).
a.
- 74 5
b.
- 19 5
c.
- 13 5
d.
1
e.
3


Solution:

(g o h)(x) = g[h(x)]
[Composition of g(x) and h(x).]

= g [x5 + 2 / 5]
[Replace h(x) = x5 + 2 / 5.]

(g o h)(- 3) = g[- 3 / 5 + 2 / 5]
[Replace x with - 3.]

= g[- 1 / 5]
[Simplify.]

= 4(- 1 / 5) - 3 = - 4 / 5 - 3
[Replace x = - 1 / 5 in g(x).]

= - 195
[Simplify.]


Correct answer : (2)
 6.  
Given f(x) = 2x4 + 5 and g(x) = x2 - 7, evaluate (g o f)(0).
a.
- 7 5
b.
18
c.
4807
d.
- 12
e.
- 35


Solution:

(g o f )(x) = g[f(x)]
[Composition of g(x) and f(x).]

= g[2x4 + 5]
[Replace f(x) = 2x4 + 5.]

(g o f )(0) = g[2(0)4 + 5]
[Replace x with 0.]

= g(5)
[Simplify.]

= (5)2 - 7 = 25 - 7 = 18
[Replace x = 5 in g(x).]


Correct answer : (2)
 7.  
Given g(x) = x3 and h(x) = 5x, evaluate (h o g)(- 2).
a.
2
b.
80
c.
- 2
d.
- 40
e.
5 4


Solution:

(h o g)(x) = h[g(x)]
[Composition of the functions h(x) and g(x).]

(h o g)(- 2) = h[g(- 2)]
[Replace x = - 2.]

= h[(- 2)3]
[Replace x = - 2 in g(x) = x3.]

= h(- 8)
[Simplify.]

= 5(- 8) = - 40
[Replace x = - 8 in h(x) = 5x.]


Correct answer : (4)
 8.  
Given f (x) = 7x - k and h(x) = x+94, find out for what value of k is ( f o h )(x) = ( h o f )(x)?
a.
- 24
b.
- 18
c.
24
d.
54
e.
18


Solution:

( f o h )(x) = f [h(x)]
[Composition of functions f (x) and h(x).]

= f [x+94]
[Replace h(x) = x+94.]

= 7(x+94) - k = 7x+634 - k
[Replace x = x+94 in f(x).]

( h o f )(x) = h[ f (x)]
[Composition of h(x) and f(x).]

= h[7x - k]
[Replace f(x) = 7x - k.]

= 7x-k+94
[Replace x = 7x - k in h(x).]

( f o h )(x) = ( h o f )(x)
[Equate ( f o h )(x) and ( h o f )(x) to find k.]

7x+634 - k = 7x-k+94

7x+63-4k4 = 7x-k+94
[Simplify.]

- 3k + 54 = 0

3k = 54 k = 18
[Solve for k.]


Correct answer : (5)
 9.  
Given h(x) = xx+1, evaluate (h o h o h)(5).
a.
5 11
b.
6 11
c.
5 6
d.
5 16
e.
11 16


Solution:

(h o h o h)(x) = (h o h) [h(x)]

= (h o h)(xx+1)
[Composition of (h o h)(x) and h(x).]

(h o h o h)(5) = (h o h)(5 / 6)
[Replace x with 5 and simplify.]

= h[h(5 / 6)]
[Composition of h(x) and h(x).]

= h[5656+1]
[Replace x = 5 / 6 in h(x) = xx+1.]

= h(5 / 11)
[Simplify.]

= 511511+1 = 5 / 16
[Replace x = 5 / 11 in h(x) = xx+1.]


Correct answer : (4)
 10.  
Which of the following is true for f (x) = 9x and h(x) = 2-xx2?
a.
(h o f)(x) = h(x)
b.
(f o f)(x) = f(x)
c.
(f o h)(x) ≠ (h o f)(x)
d.
(f o h)(x) = (h o f)(x)
e.
(f o h)(x) = f(x)


Solution:

( f o h )(x) = f [h(x)]
[Composition of f(x) and h(x).]

= f [2-xx2] = 9[2-xx2]
[Replace h(x) = 2-xx2 in f(x).]

= 18-9xx2f(x)
[Simplify.]

( h o f )(x) = h[f(x)]
[Composition of h(x) and f(x).]

= h[9x] = 2-9x81x2h(x)
[Replace f (x) = 9x and replace x = 9x in h(x).]

(f o f)(x) = f [f(x)]
[Composition of f(x) and f(x).]

= f[9x] = 9(9x) = 81xf(x)

So, ( f o h )(x) ≠ ( h o f )(x).


Correct answer : (3)

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