Question

Show that the function f: R+ [5,00) defined by f(x) = 4x² + 5 is bijective function.​

Answer 1

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

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