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Popular >

arccos(x^2)<= 1

Solution

arccos (x^{2}) \leq 1

Solution

- 1 \leq x \leq - cos (1) or cos (1) \leq x \leq 1

Interval Notation

[- 1, - cos (1)] \cup [cos (1), 1]

Decimal Notation

- 1 \leq x \leq - 0.73505 \dots or 0.73505 \dots \leq x \leq 1

Solution steps

arccos (x^{2}) \leq 1

arccos (x) \leq a

then

x \geq cos (a)

x^{2} \geq cos (1)

x^{2} \geq cos (1) : x \leq - cos (1) or x \geq cos (1)

x^{2} \geq cos (1)

For

u^{n} \geq a

, if

n

is even then

x \leq - cos (1) or x \geq cos (1)

x \leq - cos (1) or x \geq cos (1)

Domain of

arccos (x^{2}) : - 1 \leq x \leq 1

Domain definition

Find known functions domain restrictions:

- 1 \leq x \leq 1

arccos (f (x)) \Rightarrow - 1 \leq f (x) \leq 1

Solve

- 1 \leq x^{2} \leq 1 : - 1 \leq x \leq 1

- 1 \leq x^{2} \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- 1 \leq x^{2} and x^{2} \leq 1

- 1 \leq x^{2} :

True for all

x \in R

- 1 \leq x^{2}

Switch sides

x^{2} \geq - 1

If n is even,

u^{n} \geq 0

for all

u

True for all x

x^{2} \leq 1 : - 1 \leq x \leq 1

x^{2} \leq 1

For

u^{n} \leq a

, if

n

is even then

- 1 \leq x \leq 1

Combine the intervals

True for all x \in R and - 1 \leq x \leq 1

Merge Overlapping Intervals

True for all x \in R and - 1 \leq x \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

True for all

x \in R

and

- 1 \leq x \leq 1

- 1 \leq x \leq 1

- 1 \leq x \leq 1

The function domain

- 1 \leq x \leq 1

Combine the intervals

x \leq - cos (1) or x \geq cos (1) and - 1 \leq x \leq 1

Merge Overlapping Intervals

x \leq - cos (1) or x \geq cos (1) and - 1 \leq x \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

x \leq - cos (1) or x \geq cos (1)

and

- 1 \leq x \leq 1

- 1 \leq x \leq - cos (1) or cos (1) \leq x \leq 1

- 1 \leq x \leq - cos (1) or cos (1) \leq x \leq 1

Popular Examples

2sin(2x)-sqrt(3)<0

2 sin (2 x) - 3 < 0

cos(2x)<= 1

cos (2 x) \leq 1

arctan(x)-pi/2 <0.1

arctan (x) - \frac{π}{2} < 0.1

-2cos^2(x)+1>0

- 2 cos^{2} (x) + 1 > 0

sec(x)<= cos(x)

sec (x) \leq cos (x)