Recall that the smallest cosine can get is -1, and the largest is 1
I.e. the outputs are between -1 and 1, inclusive of both endpoints
This tells us the range is
This then leads directly to the fact that the domain of is .
The domain and range swap places when going from the original to the inverse function.
This swap is because x and y themselves swap roles.
Therefore, the input expression 3x+2 must be between -1 and 1
Subtracting 2 from all sides
Dividing all sides by 3
The inequality signs stay the same the entire time. They only would flip if we divided all sides by a negative number.
Answer: The domain as a compound inequality is
Verbally we can say x must be between -1 and -1/3, inclusive of both endpoints.
In interval notation, the domain is [-1, -1/3]. Don't forget the square brackets.
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