Hey I am doing test corrections and need help,
Which of the following inverse functions are defined for x=-1/2 ? Select all that apply.
Y=tan^-1 x
Y=cos^-1 x
Y=sin^-1 x
Y=csc^-1 x
Y=sec^-1 x
Y=cot^-1 x
To determine which of the given inverse functions are defined for x = -1/2, we need to check if the value of x falls within the domain of each function.
1. Y = tan^(-1)(x) (inverse tangent function)
The domain of the inverse tangent function is all real numbers. Therefore, it is defined for x = -1/2.
2. Y = cos^(-1)(x) (inverse cosine function)
The domain of the inverse cosine function is from -1 to 1 inclusive. Since x = -1/2 is within this range, the inverse cosine function is defined for x = -1/2.
3. Y = sin^(-1)(x) (inverse sine function)
The domain of the inverse sine function is also from -1 to 1 inclusive. However, x = -1/2 is not within this range, so the inverse sine function is not defined for x = -1/2.
4. Y = csc^(-1)(x) (inverse cosecant function)
The domain of the inverse cosecant function is from -∞ to -1 exclusive and from 1 to ∞ exclusive. Since x = -1/2 does not fall within these ranges, the inverse cosecant function is not defined for x = -1/2.
5. Y = sec^(-1)(x) (inverse secant function)
The domain of the inverse secant function is from -∞ to -1 exclusive and from 1 to ∞ exclusive. Since x = -1/2 does not fall within these ranges, the inverse secant function is not defined for x = -1/2.
6. Y = cot^(-1)(x) (inverse cotangent function)
The domain of the inverse cotangent function is all real numbers. Therefore, it is defined for x = -1/2.
From the given options, the inverse functions that are defined for x = -1/2 are:
- Y = tan^(-1)(x)
- Y = cos^(-1)(x)
- Y = cot^(-1)(x)
Make sure to double-check your answers with your teacher or textbook to ensure accuracy.
since |sec(x)| and |csc(x)| are both greater than 1, there is no angle x such that
sec(x) = -1/2 or csc(x) = -1/2
The rest are either always between -1 and 1 or have infinite range.