Question

23. The number of integers in the range of the function
tan [x² +1] is _ (where [.]is
x + x2 +1
greatest integer function)​

Answer 1

Answer:

Given function is, f(x)=cos

−1

[x

2

−

2

1

]+sin

−1

[x

2

+

2

1

]

For this function to be defined −1≤[x

2

+

2

1

]≤1 and −1≤[x

2

−

2

1

]≤1

or −1≤x

2

+

2

1

<2 and −1≤x

2

−

2

1

<2

or −

2

3

≤x

2

<

2

3

and −

2

1

≤x

2

<

2

5

but x

2

≥0

Taking intersection of above, domain of f(x) is x

2

∈[0,

2

3

)

Now taking different cases,

case 1. x

2

∈[0,

2

1

)

f(x)=cos

−1

[x

2

−

2

1

]+sin

−1

[x

2

+

2

1

]=cos

−1

(−1)+sin

−1

(0)=π−0=π

case 2. x

2

∈[

2

1

,

2

3

)

f(x)=cos

−1

[x

2

+

2

1

]+sin

−1

[x

2

−

2

1

]=cos

−1

(0)+sin

−1

(1)=

2

π

+

2

π

=π

Hence range of f(x) is only singleton set {π