Question 998705
Suppose f(x)=(x/8)-3 and g(x)=x^3. Find (f o g)^-1(x)
<pre>
First we find fog(x) by substituting the right side of
g(x) for x in f(x):

{{{fog(x)}}}{{{""=""}}}{{{x^3/8-3}}}

{{{fog(x)}}}{{{""=""}}}{{{x^3/8-24/8}}}

{{{fog(x)}}}{{{""=""}}}{{{(x^3-24)/8}}}

Now we find the inverse of fog(x), or fog<sup>-1</sup>(x):

Substitute y for fog(x)

{{{y}}}{{{""=""}}}{{{(x^3-24)/8}}}

Swap x and y

{{{x}}}{{{""=""}}}{{{(y^3-24)/8}}}

Solve for y.

Multiply both sides by 8

{{{8x}}}{{{""=""}}}{{{y^3-24}}}

Add 24 to both sides:

{{{8x+24}}}{{{""=""}}}{{{y^3}}}

{{{y^3}}}{{{""=""}}}{{{8x+24}}}

Take cube roots of both sides:

{{{y}}}{{{""=""}}}{{{root(3,8x+24)}}}

Replace y by fog<sup>-1</sup>(x)

fog<sup>-1</sup>(x){{{""=""}}}{{{root(3,8x+24)}}}

Edwin</pre>