Domain & Range of inverse trig functions

Domain of f⁻¹(x)

Range of the original function f(x).

Range of f⁻¹(x)

Domain of the original function f(x).

How do you find the inverse function's domain and range?

Switch the original function's domain and range.

Domain of arcsin(x)

[-1, 1]

Range of arcsin(x)

[-π/2, π/2]

Domain of arccos(x)

[-1, 1]

Range of arccos(x)

[0, π]

Domain of arctan(x)

(-∞, ∞)

Range of arctan(x)

(-π/2, π/2)

What does principal range mean?

The restricted output interval that makes an inverse function single-valued.

What are the domain and range of f⁻¹ if f has domain [2, ∞) and range [5, ∞)?

Domain: [5, ∞); Range: [2, ∞).

What are the domain and range of f⁻¹ if f has domain (-∞, 4] and range [-3, ∞)?

Domain: [-3, ∞); Range: (-∞, 4].

What are the domain and range of f⁻¹ if f has domain [-2, 6] and range [1, 9]?

Domain: [1, 9]; Range: [-2, 6].

What is the domain of √(x - 3)?

[3, ∞)

What is the range of √(x - 3)?

[0, ∞)

What are the domain and range of the inverse of √(x - 3)?

Domain: [0, ∞); Range: [3, ∞).

What is the domain of 1/(x - 2)?

(-∞, 2) ∪ (2, ∞).

What is the range of 1/(x - 2)?

(-∞, 0) ∪ (0, ∞).

What are the domain and range of the inverse of 1/(x - 2)?

Domain: (-∞, 0) ∪ (0, ∞); Range: (-∞, 2) ∪ (2, ∞).

What do square brackets [ ] mean in interval notation?

The endpoint is included; equivalent to ≤ or ≥.

What do parentheses ( ) mean in interval notation?

The endpoint is excluded; equivalent to < or >.

Can infinity ever have a square bracket?

No. Infinity is never included, so always use parentheses.

If f is one-to-one, what is true about f and f⁻¹?

Their graphs are reflections across the line y = x.

What is the domain of arccos(cos x)?

All real numbers, since cosine accepts every real input.

What is the range of arccos(cos x)?

[0, π], because arccos always returns its principal value.