Please answer these true or false trig questions! Thanks!

Question

 

1.True or False: For a trig function, y= f(x), then x=F^-1(y).Explain your answer 

2.True or False: For a one-to- one function, y=f(x), then x=f^-1(y). Explain your answer 

3.True or false: For any function, x=f^-1(y), then y=f(x). Explain your answer 

4.True or false: the graph of y=sinx is increasing on the interval [pi/2,3pi/2]. Explain your answer 

5.Explain the meaning of y=cos^-1 (x)

6.True or False. sec^-1(0.5) is undefined. Explain your answer. 

Answer 1

1. True. For any trig functions, y = sinx, cosx, tanx etc, x = arcsin y, arccos y, arctan y.

2. False. For a function to have an inverse "function", then the function must be both injective and surjective in otherwords, the function must be bijective. So, 1-1 is not enough.

3. True. Only in the case when x = (inverse) f of y is a "function".

4. False. The derivative of sin x is cos x. Apparently cos x has negative values from pi/2(exclusive) to 3 pi/2(exclusive). That means the original function sin x is decreasing on given interval. You can graph it too.

5. y = sec x. So, graph it and describe the function by seeing the graph.

6. False. cos x = 1/(sec x) = sec^(-1) (x). So, it is equivalent to saying cos(0.5) and it exists. You can punch it in calculator.