How do you find the inverse of f(x) = ln(4 - 7x) + ln(-7 - 5x)?

1 Answer
Oct 23, 2016

There is no inverse, because the quadratic formula yields two equations that returns two values for any given x and an inverse must not do this.

Explanation:

Let x = f^-1(x)

Substitute f^-1(x) everywhere there is an x:

f(f^-1(x)) = ln(4 - 7f^-1(x)) + ln(-7 - 5f^-1(x))

Use the property of inverses f(f^-1(x)) = x:

x = ln(4 - 7f^-1(x)) + ln(-7 - 5f^-1(x))

Use the identity ln(a) + ln(b) = ln(ab)

x = ln((4 - 7f^-1(x)(-7 - 5f^-1(x)))

Use the inverse of the natural logarithm on both sides:

e^x = (4 - 7f^-1(x)(-7 - 5f^-1(x))

Use the F.O.I.L. method to perform the multiplication:

e^x = -28 - 20f^-1(x) + 49f^-1(x) + 35(f^-1(x))^2

This is a quadratic.

35(f^-1(x))^2 + 29f^-1(x) - 28 - e^x = 0

At this point, we must declare that there is no inverse, because the quadratic formula yields two equations that returns two values for any given x and an inverse must not do this.

Let's proceed so that you can see the issue

Use the quadratic formula, x = {-b +-sqrt(b^2 - 4(a)(c))}/{2a}

f^-1(x) = {-29 + sqrt(29^2 - 4(35)(-28-e^x))}/{2(35)}

AND

f^-1(x) = {-29 - sqrt(29^2 - 4(35)(-28-e^x))}/{2(35)}

This is contrary to the purpose of an inverse function.

The purpose of an inverse function is, for any y returned from the f(x), give you back a single value of x that created the y; this returns two values, therefore, no inverse.