Inverse Equation and if Values exist

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In summary, the conversation discusses the concept of invertible functions in a given interval and the process of finding inverse functions. The approach to finding inverse functions is to isolate the x variable and check for the existence of positive values of x for the inverse function. However, in some cases, such as with ln(0), the inverse function may not exist. It is also suggested to look at graphs of the functions to aid in finding inverse equations.
  • #1
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]

My work process for a: let y= sech x

how do i find x in terms of y for this eq. how do i isolate the x variable from sech x.

once I find that i can see if the positive values of x for the inverse function exist from 0 to infinity.

however, i need help finding the inverse function

My work process for b.) let y= cos (ln x)

I took the expotential of both sides to get

e^y= e^(cos*(lnx))

how do i isoloate e*(lnx) to equal x , since the lnx is a part of the cosine angle expression.. i need help here...

once i find the inverse equation, i can see if y values exists for the x values from 0 to e^pie... i need help finding the inverse equation first

My work process for c.) let = y= e^(x^2)

therefore taking the natural logarithm of both sides i get

ln y= ln (e^(x^2)
ln y= x^2

therefore x= sqrt (ln y)

therefore f^-1 (x)---> inverse function is f(x)= sqrt (ln x)

when subbing (-1,2] for x to determine if y values exist,

i find the inverse function doesn't exist under those limits since ln 0 is undefined

is this the correct approach to this problem.

Please help me with these problems!
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  • #2
A function is invertible in an interval if it is "one-to-one" there: that is, that two different values of x in the interval that give the same value of y. It should be fairly obvious, for example, that (-.5)2= .52 so [itex]e^{(-.5)^2}= e^(0.5)^2[/itex].

You say at one point
ln y= x^2
therefore x= sqrt(ln y)
which is not true: if ln y= x^2 then either x= sqrt(ln y) or x= -sqrt(ln y).

Your point about ln(0) not existing is a good one.

For the others try looking at graphs of the functions.

What is an inverse equation?

An inverse equation is a mathematical equation that reverses the operation of another equation. It is used to find the input, or independent variable, when given the output, or dependent variable.

How do you solve an inverse equation?

To solve an inverse equation, you must isolate the variable that is being evaluated. This can be done by applying the inverse operation to both sides of the equation. For example, if the original equation is x + 5 = 10, the inverse operation would be subtracting 5 from both sides, resulting in x = 5.

What is the difference between an inverse equation and a reciprocal?

An inverse equation and a reciprocal are not the same thing. An inverse equation reverses the operation of another equation, while a reciprocal is the multiplicative inverse of a number. For example, the inverse of x + 5 = 10 is x - 5 = 10, while the reciprocal of 2 is 1/2.

How do you know if an inverse equation has a solution?

An inverse equation will have a solution if the function is one-to-one, meaning that each input has exactly one output. This can be determined by graphing the equation and seeing if the line passes the vertical line test.

What happens if an inverse equation does not have a solution?

If an inverse equation does not have a solution, it means that the function is not one-to-one and therefore does not have an inverse. This can happen when the equation has a repeated output for different inputs, or when there are multiple outputs for a single input.

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