# HOW TO OBTAIN THE INVERSE OF A FUNCTION

An inverse function (or anti-function) is a function that can reverse into another function. In simple words, an inverse function is a function that undoes another function.

To obtain the inverse of a function, you follow these steps

1. First, you will Substitute $y$ for $f(x) 2. Then, you will switch$x$and$y$. in other words, you put$y$anywhere you see$x$and put$y$anywhere you see$x$. 3. Then, make$y$the subject of the formula. 4. Finally, you back-substitute$f^{-1}(x)$for$y$The next five examples illustrate this steps Example 1 Given that$f(x)=x^2$. Find an expression for$f^{-1}(x)$. Solution: First, we substitute$y$for$f(x)$.$y=x^2$Next, we switch$x$and$yx=y^2$Now, let make y the subject of the formula. To achieve this, we simply add square roots to both sides$\sqrt{x}=\sqrt{y^2}y=\sqrt{x}$Now, let back-substitute$f^{-1}(x)=\sqrt{x}$Example 2 Find the$f^{-1}(x)$if$f(x)=\frac{x}{x+3}$Solution: First, we substitute$y$for$f(x)y=\frac{x}{x+3}$Next, we switch$x$and$yx=\frac{y}{y+3}$Now, let make$y$the subject of the formula$x(y+3)=yxy+3x=y3x=y-xy3x=y(1-x)\frac{3x}{1-x}=\frac{y\require{cancel}\bcancel{(1-x)}}{\require{cancel}\bcancel{1-x}}y=\frac{3x}{1-x}$Lastly, we back-substitute$f^{-1}(x)$for y$f^{-1}(x)=\frac{3x}{1-x}$Example 3 Find$f^{-1}(x)$if$f(x)=\frac{1}{x+2}$Solution: As usual, we substitute$y$for$f(x)$.$y=\frac{1}{x+2}$Now, let switch$x$and$yx=\frac{1}{y+2}x(y+2)=1xy+2x=1xy=1-2x\frac{xy}{x}=\frac{1-2x}{x}y=\frac{1-2x}{x}f^{-1}(x)=\frac{1-2x}{x}$Example 4 If$f(x)=\sqrt{x+4}$, find$f{-1}(x)$Solution:$y=\sqrt{x+4}$. Switching$x$and$yx=\sqrt{y+4}$Adding square to both sides$x^2=(\sqrt{y+4})^2x^2=y+4x^2-4=yy=x^2-4$Back-substituting$f^{-1}(x)=x^2-4$RELATED POSTS Example 5 If$f(x)=\frac{2}{x-3}-\frac{4}{x^2-4x+3}$, write an expression for$f^{-1}(x)$Solution: For simplicity, let simplify$f(x)\frac{2}{x-3}-\frac{4}{x^2-4x+3}\frac{2}{x-3}-\frac{4}{(x-3)(x-1)}$By taking the L.C.M$\frac{2(x-1)-4}{(x-3)(x-1)}\frac{2x-2-4}{(x-3)(x-1)}\frac{2x-6}{(x-3)(x-1)}\frac{2\require{cancel}\bcancel{(x-3)}}{\require{cancel}\bcancel{(x-3)}(x-1)}\frac{2}{x-1}$Therefore, the above function can be written as:$f(x)=\frac{2}{x-1}$Having simplified the function, let's now find the inverse function$y=\frac{2}{x-1}$Now, let's switch$y$and$xx=\frac{2}{y-1}x(y-1)=2xy-x=2xy=2+x\frac{xy}{x}=\frac{2+x}{x}y=\frac{2+x}{x}f^{-1}(x)=\frac{2+x}{x}\$

There you have it!.We will be evaluating composite function in my next post. Meanwhile, If you have got questions relating to this, do well to ask in the comment box.

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