Having trouble with true/false questions in Trigonometry. They read as follows -
True or False: For a trigonometric function, y=f(x), then x=F^-1(y). Explain your answer.
True or False: For a one to one function, y=f(x), then x=f^-1(y). Explain your answer.
True or False: For any function, x=f^-1, then y=f(x). Explain your answer.
In the lesson, we were taught that for inverses, capitals, F, denote inverse functions, and lowercase, f, denote inverse relations. From the info taught to me, I have these answers:
1 - False (The domains and ranges of inverse trig functions, F, are restricted)
2 - True (Since the function is one-to-one, there is no chance of the original function going over the allowable domain of the inverse)
3 - I believe that this is true, since x = f^-1(y) can be undone by the original.
X=F^1(y)
You have made good observations regarding the true/false questions in Trigonometry. Let's go through each question and explain the answers, as you have suggested:
1. True or False: For a trigonometric function, y=f(x), then x=F^-1(y).
Your answer: False.
Explanation: In general, for trigonometric functions, the inverse function is denoted as "arcsin", "arccos", "arctan", etc. So, if y = f(x) is a trigonometric function, the inverse function is not denoted as F^-1(y), but rather as f^-1(y). It's important to use the appropriate notation when referring to inverse trigonometric functions.
2. True or False: For a one-to-one function, y=f(x), then x=f^-1(y).
Your answer: True.
Explanation: If a function is one-to-one (injective), it means that each element in the domain maps uniquely to an element in the range. In such cases, if y = f(x), then x = f^-1(y). The notation f^-1(y) represents the inverse function that undoes the original function, allowing you to retrieve the original value of x.
3. True or False: For any function, x=f^-1(y), then y=f(x).
Your answer: True.
Explanation: If x = f^-1(y), it means that the inverse function f^-1 reverses the original function f. So, if you substitute x back into the original function, you get y = f(x). This is a fundamental property of inverse functions.
Remember, your understanding of the concepts and their notation is essential in correctly answering these true/false questions. Keep practicing and working with the proper notation in trigonometry, and you'll become more confident in solving these types of problems.