Inverse functions, and limits help

[johnq2k7]

Please help me with these following problems:

1.)Indicate whether each of the following functions is invertible in the given interval. Explain
a.) sech x on [0,infinity)

b.) cos (ln x) on (O, e^pie]

c.) e^(x^2) on (-1,2]


2.) Evaluate the following limits, justifying your answers. If a limit does not exist explain why.

a.) lim (x--> inf.) (3x^3 +cos x)/(sin x- x^3)
b.) lim (x-->Pie(+)) (tan^-1 (1/(x-Pie)))/(Pie-x)
c.) lim (x--> 0(+)) (sqrt(x+sin x))(ln x)
d.) lim (x--> 1(-)) (cos^-1(x))/(1-x)

3.) give the equation of the line tangent to the curve at the given point.
a.) (y)(tan^-1 x)= x*y at (sqrt(3),0)
b.) ln y= x^2 +(2)*e^x at (0, e^2)


My attempt at a solution for a.)

1.) sech x on [0,infinity)

let y= sech x
therefore, y*sech^-1 x= sech^-1 x* sech x

therefore x= y*sech^-1 x

since it is an inverse function switch x and y variables

therefore f^-1(x)= x*sech^-1 (y)

how do i determine, it's intervible in [0, infinity)... I'm not sure if this prior work is correct

 

[quantumdude]

1.) sech x on [0,infinity)

let y= sech x
therefore, y*sech^-1 x= sech^-1 x* sech x

therefore x= y*sech^-1 x

since it is an inverse function switch x and y variables

therefore f^-1(x)= x*sech^-1 (y)

That won't do. You've simply assumed the existence of sech^-1(x). You need to show that the function is 1-1 on $[0,\infty)$ (that is, you need to show that it passes the so-called horizontal line test on that interval). One way to do that is to show that the function is monotonic on that interval. Have you been taught a theorem that shows you how to determine the monotonicity of a function?

 

[johnq2k7]

yes, basically for the horizontal line test, each x values only one unique f(x) value if it is to be one-to-one...

since, sech(x) = 1 / cosh(x) = 2 / (e^x + e^-x)

can i compute y= 2/(e^x +e^-x)

and solve for x:

than switch the variables for y and x to find the inverse function?

and then plug in the values of [0,infinity) into x

and determine if a f(x) value exists?

is this a proper method

 

[quantumdude]

In principle that's fine, but in practice it's a pain. It would be easier to show that y=sech(x) is monotonic (that is, either always increasing or always decreasing). Now the question for you is, how do you tell whether a function is increasing or decreasing on an interval?

 

[johnq2k7]

but wouldn't I have to determine the graph of the inverse of the sech (x) equation in order to determine if a value exists for [0, infinity)

 

[quantumdude]

You don't have to find the inverse, you just have to determine whether or not an inverse exists. There's a big difference. You need to use the following facts.

1.) A function $f:D\longrightarrow R$ has an inverse $f^{-1}:R\longrightarrow D$ if and only if $f$ is 1-1 and onto. The onto part is easy: just assume that the output values of the function are the range of the function. It's the 1-1 part that you need to worry about. This brings is to...

2.) A function is 1-1 on its domain if and only if it is monotonic on its domain.

So how do you show that $f$ is monotonic?