Problem 6 : A continuous function
f
:
R
→
R
f
:
R
→
R
is given by :

f
x
=
x
2
+
4
x
+
30
x
2
−
8
x
+
18
f
x
=
x
2
+
4
x
+
30
x
2
−
8
x
+
18

Determine increasing or decreasing nature of the function and check whether function is an injection?

Solution :

Statement of the problem : The rational function is a continuous function. Hence, we can determine its increasing or decreasing nature in its domain by examining derivative of the function.

⇒
f
′
x
=
x
2
−
8
x
+
18
2
x
+
4
−
x
2
+
4
x
+
30
2
x
−
8
x
2
−
8
x
+
18
2
⇒
f
′
x
=
x
2
−
8
x
+
18
2
x
+
4
−
x
2
+
4
x
+
30
2
x
−
8
x
2
−
8
x
+
18
2

⇒
f
′
x
=
−
12
x
2
+
2
x
−
26
x
2
−
8
x
+
18
2
⇒
f
′
x
=
−
12
x
2
+
2
x
−
26
x
2
−
8
x
+
18
2

The denominator is a square of a quadratic expression, which evaluates to a positive number. On the other hand, the discreminant of the quadratic equation in the numerator is :

⇒
D
=
2
2
−
4
X
1
X
−
26
=
4
+
104
=
108
⇒
D
=
2
2
−
4
X
1
X
−
26
=
4
+
104
=
108

It means that derivative has different signs in the domain interval. Therefore, the function is a combination of increasing and decreasing nature in different intervals composing domain. Thus, function is not monotonic in the domain interval. Hence, we conclude that function is not an injection.