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Properties of Function Worksheet

Properties of Function Worksheet
  • Page 1
 1.  
Choose the graph which represents a function.


a.
Graph -A
b.
Graph -B
c.
Graph -C
d.
Graph -A & Graph -C


Solution:

A vertical line intersects the graph -A in only one point , so the given curve is the graph of a function.
[Vertical line test.]

A vertical line intersects the graph -B in two popints , so the given curve is not the graph of a function.
[Vertical line test.]

A vertical line intersects the graph-C in two points, so the given curve is not the graph of a function.
[Vertical line test.]


Correct answer : (1)
 2.  
What is the domain of the function defined by h(x) = 1x + 4x - 4 ?
a.
(- ∞, ∞)
b.
(- ∞, 0) U (4 , ∞)
c.
(- ∞, 0) U (0, 4 ) U (4 , ∞)
d.
(- ∞, 0] U [0, 4 ] U [4 , ∞)


Solution:

h(x) = 1x + 4x - 4 = 5x - 4x(x - 4)

x ≠ 0 and x ≠ 4.
[The denominator of a fraction must not be zero.]

The domain is (-∞, 0) U (0, 4) U (4, ∞).


Correct answer : (3)
 3.  
Find the domain of the function defined by g(x) = 36 - x(x + 2)(x2 + 2).
a.
(- ∞, - 2 ] U [- 2 , 36]
b.
(- ∞, - 2 ) U (- 2 , 36]
c.
[- ∞,- 2 ) U (- 2 , 36]
d.
(- ∞, - 2 ) U (- 2 , ∞)


Solution:

g(x) = 36 - x(x + 2)(x2 + 2)

x + 2 ≠ 0 and x ≠ - 2.
[The denominator of a fraction must not be zero.]

36 - x ≥ 0
[The expression under a radical must not be negative.]

x ≤ 36
[Solve.]

The domain is (-∞, - 2) U (- 2, 36].


Correct answer : (2)
 4.  
A relation f, which associates to each element of set A with a unique element of set B is called
a.
A mapping from AA
b.
A function from B A
c.
Relation
d.
Function from AB


Solution:

A relation f , which associates to each element of set A with a unique element of set B is called Function from AB .
[Definition.]


Correct answer : (4)
 5.  
If f is a function from set A to set B, then which of the following is correct?
a.
A is called domain of f and B is called codomain of f
b.
A is called range of f and B is called domain of f
c.
A is called codomain of f and B is called domain of f
d.
A is called domain of f and B is called range of f


Solution:

If f is a function from set A to set B then A is called domain of f and B is called codomain of f.


Correct answer : (1)
 6.  
Which of the following formulas represents a function?
a.
y = x - 4, [4, ∞)
b.
y = x2± 3
c.
y = x + 5
d.
Both A and C


Solution:

For the choice A: y = x - 4 on the interval [4,∞)
Here y is a function of x. Because, the graph of the given formula intersects the vertical line at only one point with x-coordinate 4. Hence, the given formula represents a function.
[Vertical line test.]

For the choice B: y = x2 ± 3
For unique value of x, there exists two values for y (or) the graph of the given equation intersects vertical line at more than one point with x-coordinate 0. Hence, the given formula does not represent a function.
[Vertical line test.]

For the choice C: y = x + 5
The graph of the given equation intersects vertical line at only one point with x- coordinate 0. Hence, the given formula represents a function.
[Vertical line test.]


Correct answer : (4)
 7.  
Choose the formula which is not a function.
a.
x = 2 + y
b.
y = x + 5
c.
y = x2
d.
x = 2 y2


Solution:

For the choice A: x = 2 + y y = 2 - x
The graph has unique y value for a unique value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice B: y = x + 5
The graph has unique y value for a unique value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice C: y = x2
Here, the graph has two y values for every positive value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice D: x = 2y2 y = ± x2
Here, the graph has two y values for every positive value of x. Hence, it is not a function. In other words, the vertical line intersects the graph in more than one point.
[Vertical line test.]


Correct answer : (4)
 8.  
Find the domain of the function defined by f (x) = x + 3.
a.
(- ∞, ∞)
b.
(- ∞, 3 ]
c.
[- 3 , ∞]
d.
[ - 3 , ∞)


Solution:

f(x) = x + 3

x + 3 ≥ 0.
[The expression under a radical sign should be positive.]

x ≥ - 3
[Solve.]

Hence, the domain of f is the interval [- 3, ∞).


Correct answer : (4)
 9.  
What is the domain of the function f (x) = 7x2?
a.
[- ∞, 0) (0, ∞)
b.
( - ∞, 0) [ 0, ∞)
c.
(- ∞, 0) (0, ∞)
d.
(- ∞, 0] (0, ∞)


Solution:

f (x) = 7x2

0 is the only number, which is not in the domain of f (x).
[For all values of x except 0, 7x2 is defined.]

Hence, the domain of f (x) is (- ∞, 0) (0, ∞).


Correct answer : (3)
 10.  
Find the range of the function f(x) = 9x3
a.
[ - ∞, ∞]
b.
(- ∞, ∞)
c.
(- ∞, 0]
d.
(- ∞, 0) U (0,∞)


Solution:

The domain of f(x) = 9x3 is (- ∞, ∞).

For all the real values of x, 9x3 is a real number or 9x3 (- ∞, ∞).

So, the range of f(x) = 9x3 is (- ∞,∞).


Correct answer : (2)

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