If a function f is defined by formula f(x)=√(1+x3) and g is an inverse function of f, find g

Question

If a function f is defined by formula f(x)=√(1+x³) and g is an inverse function of f, find g

Answer 1

Generally speaking, to find the inverse of a function you simply change f(x) to just x and all other x's to y. Then solve for y.

In this case, f(x)=√(1+x³) --> x=√(1+y³). Use the reverse of the Order of Operations to solve for y. Then change y to g(x).

The catch, of course, is that your f(x) is a square root function. So you must restrict the domain on the inverse for all values to work.

Be sure to check your result by plugging g(x) into f(x) - substitute all of g(x) into each x in f(x) - and simplifying. If your result is "x", your inverse is correct.

ie, if f(x)=3x-2 & g(x)=(x+2)/3...

then 3((x+2)/3)-2 = x+2-2 = x --> inverses

however, if f(x)=3x-2 & g(x)=x/3 + 2

then 3(x/3+2)-2 = x+6-2 = x+4 --> not inverses

I hope this helps. Please let us know if you need further assistance.

Answer 2

x-->x^3-->x^3+1-->sqrt(x^3+1) = f(x), x>-1

Now go in reverse and start with x

x-->x^2-->x^2-1--> cuberoot(x^2-1) = g(x)