Question 381052
To find an inverse:<ol><li>If function notation is present, replace it with a "y".</li><li>Rewrite the equation "swapping" the x's and y's. IOW, where there is an "x" replace it with a "y" and where there is a "y" replace it with an "x". This actually changes the equation. The new equation is the inverse relation for the previous equation. But it is not in the desired form, yet.</li><li>Solve the inverse equation for y, if possible.<ul><li>If it is possible to solve for y and if function notation was used initially, then replace the "y" with f<sup>-1</sup>(x).</li><li>If it is not possible to solve for y, then the inverse relation is not a function.</li></ul></li></ol>
Let's see this in action:
{{{f(x)=-2+log(4, (x-3))}}}
1) Replace f(x) with y:
{{{y=-2+log(4, (x-3))}}}
2) Swap the x's and y's:
{{{x=-2+log(4, (y-3))}}}
3) Solve for y:
{{{x + 2 = log(4, (y-3))}}}
Rewrite in exponential form:
{{{4^(x + 2) = y-3}}}
{{{4^(x + 2)+3 = y}}}
We solved for y and we have function notation initially so we will replace y:
{{{4^(x + 2)+3}}} = f<sup>-1</sup>(x)