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Function Notation Worksheet

Function Notation Worksheet
  • Page 1
 1.  
Which of the following is true about the given functions f (x) = x2+x-2x-1 and g (x) = x + 2?
a.
Both functions f(x) = g(x) + 1
b.
Both functions f(x), g(x) are identical
c.
Both functions f(x), g(x) are not identical
d.
Both functions f(x), g(x) have the same domain


Solution:

f (x) = x2+x-2x-1 and g (x) = x + 2

Clearly, the function f (x) is continuous for all values of x except at x = 1. Hence, the domain of f(x) is R - {1}.

For all real x except 1, f (x) = x + 2
[Use polynomial division.]

The function g (x) is defined for all real values of x. Hence, the domain of g(x) is R.

So, f (x) = x + 2 with domain R - {1}, and g(x) = x + 2 with domain R.

Though both the functions have the same definitions , they are not identical as they have different domains.


Correct answer : (2)
 2.  
Choose the domain of the function f (x) = 1x + 16.
a.
All real x except - 16
b.
All real x except 16
c.
All real vaues
d.
All real x except - 17


Solution:

The domain of the function f (x) = 1x + 16 is all the real values of x except - 16
[f (x) = 1x + 16 is defined for all the real values of x except -16.]


Correct answer : (1)
 3.  
Which of the following transformations have to be applied to the graph of reciprocal function f (x) = 1x to get the graph of g (x) = 2x + 6?
a.
Translation to the left by 2 units and a vertical stretch by a factor of 6.
b.
Translation to the left by 6 units and a vertical stretch by a factor of 6.
c.
Translation to the left by 6 units and a vertical stretch by a factor of 2.
d.
Translation to the left by 2 units and a vertical stretch by a factor of 2.


Solution:

f (x) = 1x and g (x) = 2x + 6

g (x) = 2x + 6

= 2 (1x + 6)

= 2 · f (x + 6) where f (x) = 1x

The graph of g is the graph of the reciprocal function f translated to the left by 6 units then stretched vertically by a factor of 2.


Correct answer : (3)
 4.  
Which of the following transformations have to be applied to the graph of reciprocal function f (x) = 1x to get the graph of g (x) = 5x - 29x - 6?
a.
Translation to the right by 5 unit and then translation vertically up by 6 units.
b.
Translation to the right by 6 unit and then translation vertically up by 5 units.
c.
Translation to the right by 6 unit and then translation vertically up by 6 units.
d.
Translation to the right by 5 unit and then translation vertically up by 5 units.


Solution:

f (x) = 1x and g (x) = 5x - 29x - 6

g (x) = 5x - 29x - 6

g (x) = 5 + 1x - 6
[Use polynomial division.]

So, g (x) = 5 + f (x - 6) Where f (x) = 1x.

The graph of g is the graph of the reciprocal function f translated to the right by 6 unit and then translated up by 5 units.


Correct answer : (2)
 5.  
Which of the following is the vertical asymptote of h(x) = x2+36x?
a.
y = 0
b.
x = - 36
c.
x = 36
d.
x = 0


Solution:

f (x) = x2+36x

x = 0
[Find the zeros of the denominator.]

The vertical asymptote is the line x = 0.
[For the graph of a rational function vertical asymptotes occur at the zeros of the denominator, which are not the zeros of the numerator.]


Correct answer : (4)
 6.  
Find the x - intercept of the graph of f (x) = x2-9x+20x+5.
a.
(4 , 0) and (- 5, 0)
b.
(- 4 , 0) and (- 5, 0)
c.
(- 4, 0) and (5, 0)
d.
(4 , 0) and (5, 0)


Solution:

f (x) = x2-9x+20x+5

x2 - 9x + 20 = 0
[Solve the numerator for the zeros.]

(x - 5)(x - 4) = 0
[Factor.]

x - 5 = 0 or x - 4 = 0

The zeros of the numerator are x = 4 and 5

So, the x - intercepts of the graph of the given function are (4, 0) and (5, 0).
[The x - intercepts of the graph of a rational function are the zeros of its numerator.]


Correct answer : (4)
 7.  
Which of the following would be the end behavior asymptote of y = x2+25x - 5?
a.
y = x + 25
b.
y = x - 5
c.
y = x + 5
d.
y = 5x


Solution:

y = x2+25x - 5

y = (x + 5) + 50x - 5
[Use polynomial division.]

Hence, the end behaviour asymptote of the graph of the given function is y = x + 5
[The end behaviour asymptote of y = q (x) + r(x)g(x) is y = q (x).]


Correct answer : (3)
 8.  
Which of the following transformations have to be applied to the graph of reciprocal function f (x) = 1x to obtain the graph of g (x) = 88 - 9xx - 10?
a.
Translation to the right by 10 units followed by a reflection about y-axis.
b.
Translation to the right 10 units, vertical stretch by a factor of 10, a reflection across x - axis and translation vertically down by 9 units.
c.
Translation to the right 10 units, vertical stretch by a factor of 2 , a reflection across x - axis and translation vertically down by 9 units.
d.
Translation to the right 10 units, vertical stretch by a factor of 2 , a reflection across x - axis and translation vertically down by 2 units.


Solution:

f (x) = 1x and g (x) = 88 - 9xx - 10

g (x) = 88 - 9xx - 10

g(x) = - 9 - 2x - 10
[Use polynomial division.]

= - 9 - 2(1x - 10)

= - 9 - 2 f (x - 10) where f (x) = 1x

The graph of g(x) can be obtained by a translation to the right 10 units, vertical stretch by a factor of 2, a reflection across the x - axis and then a translation vertically down by 9 units on the graph of f(x).


Correct answer : (3)
 9.  
Use the graph of the function f (x) to evaluate limx f (x).


a.
3
b.
c.
-∞


Solution:

As the value of x is increasing without any bound, the value of f (x) is approaching the zero.
[From the graph.]

Hence, limx f (x) = 0.


Correct answer : (1)
 10.  
Use the graph of the function f (x) to evaluate limx-2- f(x).

a.
-∞
b.
c.
3


Solution:

Evidently, from the graph, the curve is discontinuous at x = -2.

As x -2-, the value of f (x) is approaching a very large value (∞).
[From the graph.]

Hence, limx-2- f(x) = ∞.


Correct answer : (3)

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