Question 258498
To find the inverse of a function:<ol><li>Repalce the function notation with "y"</li><li>Swap the x's and y's in your equation. This swapping of the x and y changes to equation from the equation of the function to the equation of the inverse.</li><li>Solve the inverse equation for y, if possible.</li><li>If successful in solving the inverse for y, then the inverse is a function, too. Replace the y with the notation for inverse functions.</li></ol>
Let's see how this works:
{{{f(x) = 64x^3}}}
1. Replace the function notation with "y":
{{{y = 64x^3}}}
2. Swap the x's and y's:
{{{x = 64y^3}}}
This is the equation for the inverse relation.
3. Solve the inverse for y, if possible. We'll start by dividing both sides by 64:
{{{x/64 = y^3}}}
Next we'll find the cube root of each side:
{{{root(3, x/64) = root(3, y^3)}}}
Using the property of radicals, {{{root(a, p/q) = root(a, p)/root(a, q)}}}, on the left side and simplifying the cube root on the right side we get:
{{{root(3, x)/root(3, 64) = y}}}
Since {{{root(3, 64) = 4}}}, this simplifies to:
{{{root(3, x)/4 = y}}}
4. If successful in solving for y, then replace the y with the notation for inverse functions:
{{{root(3, x)/4 = f^(-1)(x)}}} or {{{(1/4)root(3, x) = f^(-1)(x)}}}<br>
(NOTE: Algebra.com's software mistakenly puts a multiplication symbol in the inverse function notation. This is a mistake. There is no multiplication involved.)