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 $y$

$x=y^2$

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 $y$

$x=\frac{y}{y+3}$

Now, let make $y$ the subject of the formula

$x(y+3)=y$

$xy+3x=y$

$3x=y-xy$

$3x=y(1-x)$

$\frac{3x}{1-x}=\frac{y\require{cancel}\bcancel{(1-x)}}{\require{cancel}\bcancel{1-x}}$

$y=\frac{3x}{1-x}$

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 $y$

$x=\frac{1}{y+2}$

$x(y+2)=1$

$xy+2x=1$

$xy=1-2x$

$\frac{xy}{x}=\frac{1-2x}{x}$

$y=\frac{1-2x}{x}$

$f^{-1}(x)=\frac{1-2x}{x}$

Example 4

If $f(x)=\sqrt{x+4}$, find $f{-1}(x)$

Solution:

$y=\sqrt{x+4}$.

Switching $x$ and $y$

$x=\sqrt{y+4}$

Adding square to both sides

$x^2=(\sqrt{y+4})^2$

$x^2=y+4$

$x^2-4=y$

$y=x^2-4$

Back-substituting

$f^{-1}(x)=x^2-4$

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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 $x$

$x=\frac{2}{y-1}$

$x(y-1)=2$

$xy-x=2$

$xy=2+x$

$\frac{xy}{x}=\frac{2+x}{x}$

$y=\frac{2+x}{x}$

$f^{-1}(x)=\frac{2+x}{x}$

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.

Alternatively, you can ask our telegram community.


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