Precalculus : Graphs and Inverses of Trigonometric Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #4 : Graphing Inverse Trig Functions

State the domain and range of  and .

Possible Answers:

 domain: 

 range: 

 domain: 

 range: , such that 

 domain: , such that 

 range: 

 domain: 

 range:  

 domain: 

 range: 

domain: 

 range:  

 domain: 

 range: 

 domain: 

 range:  

 domain: 

 range: 

 domain: 

 range: 

Correct answer:

 domain: , such that 

 range: 

 domain: 

 range:  

Explanation:

The two graphs help provide a visual. The domain is all possible x values, and the range is all possible y values. The arctan function exchanges all x and y values of the tangent function. However, in order to continue being a function (by passing the horizontal line test), the arctangent function's range is restricted to . It is not a coincidence that the range of the tangent function is equal to the domain of the arctangent function because of the nature of inverse relationships.

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 domain: , such that 

 range: 

 domain: 

 range:  

You can see that while the range and the  domain match, the  domain and the  range are not exact matches; this is so that  can be a function (by being restricted).

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