Show that the function f:N --> y: f(x) = 4 x ² + 12x + 15 is an onto function
Answers
Step-by-step explanation:
12th
Maths
Relations and Functions
Types of Functions
Let f: N → Y be a function ...
MATHS
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Asked on December 26, 2019 by
Mishthi D'Souza
Let f:N→Y be a function defined by f(x)=4x
2
+12x+15, where Y= range of f. Show thatf is invertible and find the inverse of f.
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ANSWER
Solution:-
f:N→Y
f(x)=4x
2
+12x+15
A function is invertible if the function is one-one and onto.
Let x
1
,x
2
∈N, such that
f(x
1
)=f(x
2
)
4x
1
2
+12x
1
+15=4x
2
2
+12x
2
+15
⇒(x
1
2
−x
2
2
)+3(x
1
−x
2
)=0
⇒(x
1
−x
2
)(x
1
+x
2
+3)=0
∵x
1
,x
2
∈N
⇒x
1
+x
2
+3
=0
∴x
1
=x
2
Thus, f(x) is one-one.
The function is onto if there exist x in N such that f(x)=y
∴4x
2
+12x+15=y
⇒4x
2
+12x+(15−y)=0
Here,
a=4
b=12
c=15−y
From quadratic formula, x=
2a
−b±
b
2
−4ac
, we have
x=
2×4
−12±
12
2
−4×4×(15−y)
⇒x=
2
−3±
y−6
∵x∈N
∴x=
2
−3+
y−6
f
−1
(x)=
2
−3+
x−6
;x≥6