Inverse tan domain and range | Trigonometry | Khan Academy

TL;DR

Find the inverse of the given function using the inverse tangent and determine its domain.

Transcript

Voiceover:We're told given g of x is equal to ten of x minus three pi over two plus six, find the g inverse of x. They want us to type that in here and then they also want us to figure out what is the domain of g inverse, the domain of g inverse of x. I've got my little scratch pad here to try to work that through. Let's figure out what g inverse o... Read More

Key Insights

• ☺️ Swapping x with the inverse function of x and the function with x can help find the inverse of a function.
• ❓ The inverse of a function can be found by solving for the inverse using inverse trigonometric functions.
• #️⃣ The domain of the inverse function is usually all real numbers unless there are specific restrictions.
• ❓ The tangent function has a domain of -π/2 to π/2 to ensure it can be inverted by restricting multiple mappings to the same output.
• 🧡 The range of the inverse tangent function is -π/2 to π/2, excluding the boundaries.

Questions & Answers

Q: How do you find the inverse of a function?

To find the inverse of a function, replace x with the inverse function of x and the function itself with x, then solve for the inverse function of x.

Q: What is the inverse of g(x) = tan(x - 3π/2 + 6)?

The inverse of g(x) is g inverse of x = arctan(x - 6) + 3π/2.

Q: What is the domain of the inverse function g inverse of x?

The domain of g inverse of x is -∞ to ∞, meaning it can take any real number as input.

Q: Why is the domain of the tangent function restricted to -π/2 to π/2?

The domain of the tangent function is restricted to -π/2 to π/2 to make it invertible and prevent multiple elements of the domain from mapping to the same element of the range.

Summary & Key Takeaways

• The video explains how to find the inverse of a function and determine its domain using the example of g(x) = tan(x - 3π/2 + 6).

• To find the inverse, replace x with g inverse of x and g(x) with x, then solve for g inverse of x.

• The inverse of g(x) is given by g inverse of x = arctan(x - 6) + 3π/2.

• The domain of g inverse of x is -∞ to ∞, meaning it can take any real number as input.