Inverse Functions of Hyperbolic Cosine

The video tutorial explains how to find the logarithmic form of the inverse function of cosh x. It begins by swapping x and y, then derives the equation by multiplying and rearranging terms. The process involves completing the square and handling the plus-minus sign. The inverse function is defined by restricting the domain to ensure a one-to-one function. Finally, the inverse function is expressed in logarithmic form, concluding with the natural log of x plus the square root of x squared minus one.

What is the first step in finding the inverse function of cosh x?

• Swap x and y

• Complete the square

• Multiply both sides by 2

• Take the logarithm of both sides

After swapping x and y, what expression do we get for x?

• x = (e^y - e^-y) / 2

• x = (e^y + e^-y) / 2

• x = e^y + e^-y

• x = e^y - e^-y

What operation is performed after multiplying by 2 in the derivation?

• Multiply by e^y

• Divide by e^y

• Subtract x from both sides

• Add 1 to both sides

What is the purpose of completing the square in this derivation?

• To simplify the equation

• To find the roots of the equation

• To isolate e^y

• To eliminate the square root

Why is the domain of cosh x restricted to x ≥ 0?

• To eliminate negative values

• To simplify the equation

• To ensure the function is one-to-one

• To ensure the function is many-to-one

What is the result of taking the logarithm of both sides in the final derivation?

• It simplifies the equation

• It eliminates the square root

• It isolates x

• It provides the inverse function

What symmetry property is used in the final derivation?

• No symmetry is used

• Symmetry about the origin

• Symmetry about the y-axis

• Symmetry about the x-axis

What is the final expression for the inverse function of cosh x?

• ln(x^2 - 1)

• ln(x + sqrt(x^2 - 1))

• ln(x - sqrt(x^2 - 1))

• ln(x + 1)

What happens to the negative solution in the final expression?

• It is used in the final result

• It is discarded

• It is added to the positive solution

• It is multiplied by the positive solution

Why is the negative solution discarded in the final expression?

• It is not a real number

• It is not needed for symmetry

• It complicates the equation

• It is not valid for x ≥ 0