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Functions and Inverse Functions Worksheet

Functions and Inverse Functions Worksheet
  • Page 1
 1.  
Which of the following is the graph of a relation, that has an inverse, which is a function?

a.
Graph B
b.
Graph C
c.
Graph A
d.
None of these


Solution:

The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.
[Horizontal line test.]

For the choice A: Since, the graph intersects the horizontal line at more than one point, hence it fails in the horizontal line test.

For the choice B: Since, the graph intersects the horizontal line at more than one point, hence it fails in horizontal line test.

For the choice C: Since, the graph intersects the horizontal line at one point, hence it passes through horizontal line test.

So, graph -C represents the relation that has an inverse, which is a function.


Correct answer : (2)
 2.  
Choose the graph of the inverse function for the graph shown.


a.
Graph-A
b.
Graph-B
c.
Graph-C
d.
None of the above


Solution:

If we choose any point (a, b) arbitrarily on the given graph of the function then the point (b, a) must lie on the graph of its inverse function.
[The inverse reflection principle.]

Take the given graph and choose a point (2, 1.25).
[Read the value of y at x = 2 from the graph carefully.]

Here we choose (2, 1.25) arbitrarily on the given graph of the function then the point (1.25, 2) must lie on the graph of it’s inverse function.

For the graph - A:
At x = 1.25 the value of y is approximately 1.1. So, (1.25, 1.1) is the point on the graph.
Hence, it is not the graph of the inverse function of the given graph.
[From the graph - A.]

For the graph - B:
At x = 1.25 the value of y is - 2. So, (1.25, - 2) is the point on the graph.
Hence, it is not the graph of the inverse function of the given graph.
[From the graph - B.]

For the graph - C:
At x = 1.25 the value of y is 2. So, (1.25, 2) is the point on the graph.
Hence, it is the graph of the inverse function of the given graph.
[From the graph - C.]


Correct answer : (3)
 3.  
Which of the ordered pairs are in the relation 3x + 4y = 8?
a.
(1, 1)
b.
(3, - 2)
c.
(4, - 1)
d.
(- 2, 3)


Solution:

3x + 4y = 8

3(1) + 4(1) = 7 ≠ 8
For (1, 1),
[Substitute x = 1, y = 1.]

3(3) + 4(- 2) = 1 ≠ 8
For (3, - 2),
[Substitute x = 3, y = - 2.]

3(4) + 4(- 1) = 8 = 8
For (4, - 1),
[Substitute x = 4, y = - 1.]

So, (4, - 1) is in the relation defined by 3x + 4y = 8.


Correct answer : (3)
 4.  
If f and g are two functions, then the function (f + g) is defined only when
a.
Domain of f ≠ Domain of g
b.
Both f and g have intersecting domains
c.
The new function (f + g) has the domain of g only
d.
The new function (f + g) has the domain of f only


Solution:

If f and g are two functions, then the function (f + g) is defined only when both f and g have intersecting domains.


Correct answer : (2)
 5.  
If f(x) = x2 and g(x) = x + 7, then find (f + g).
a.
x2 + x - 7 
b.
x2 - x + 7 
c.
x2 + x + 7 
d.
x + x2 + 7 


Solution:

f(x) = x2 and g(x) = x + 7

(f + g) (x) = f(x) + g(x)
[Sum rule of functions.]

= x2 + x + 7


Correct answer : (3)
 6.  
If two functions are defined by f(x) = 6x + 2 and g(x) = x + 9, then find gof(2).
a.
24
b.
14
c.
23
d.
9


Solution:

f(x) = 6x + 2 and g(x) = x + 9

gof(2) = g[f(2)]
[Use the definition of composite function.]

= g[6(2) + 2]
[Substitute x = 2 in f(x) = 6x + 2.]

= g[14]

= 14 + 9
[Substitute x = 14 in g(x) = x + 9.]

= 23


Correct answer : (3)
 7.  
If two functions are defined by f(x) = x2 - 12 and g(x) = x + 6, then what is (fog)(x)?
a.
x - 6
b.
x + 6
c.
6 - x
d.
x + 6


Solution:

f (x) = x2 - 12 and g (x) = x + 6

(fog)(x) = f(g(x))
[Use the definition of composite function.]

= f (x + 6)
[Substitute in g(x) = x + 6.]

= (x + 6)2 - 12
[Substitute in f(x) = x2 - 12.]

= x - 6


Correct answer : (1)
 8.  
If two functions are defined by f(x) = x2 - 3 and g(x) = 1x - 9, then write the domain of the composite function (fog)(x).
a.
R - {9}
b.
(-1, ∞)
c.
(0, ∞)
d.
R


Solution:

f(x) = x2 - 3 and g(x) = 1x - 9

(fog)(x) = f(g(x))
[Use the definition of composite function.]

= f(1x - 9) = 1(x - 9)2 - 3
[Use g(x) = 1x - 9, f(x) = x2 - 3.]

The function (fog)(x) is defined x R except at x = 9. Hence, the domain of the composite function is R - {9}.


Correct answer : (1)
 9.  
If the functions are defined by f(x) = x + 5 and g(x) = | x + 3 |, then what is the domain of the function fg?
a.
[- 5, - 3] ∩ (- 3, ∞)
b.
(- ∞, ∞)
c.
(- 5, - 3) ∩ [ - 3, ∞)
d.
[ - 5, - 3) (- 3, ∞)


Solution:

f(x) = x + 5 and g(x) = | x + 3 |

(fg)(x) = f(x)g(x)
[Use the quotient rule of functions.]

= x + 5|x + 3|
[Use f(x) = x + 5 and g(x) = | x + 3 |.]

The composite function is not defined when x = - 3 and x < - 5.

Hence, the domain is [ - 5, - 3) (- 3, ∞).


Correct answer : (4)
 10.  
Choose the direct relationship between x and y when x = t6 and y = t - 35, where t is a parameter.
a.
y = (x + 35)6
b.
x = (y + 35)6
c.
x = (y - 35)6
d.
y = (x - 35)6


Solution:

x = t6 and y = t - 35

t = y + 35
[Write t in terms of y.]

x = (y + 35)6
[Substitute t = y + 35 in x = t6.]


Correct answer : (2)

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