What are inverse functions?

Let $$X$$ and $$Y$$ be two sets, and $$f$$ be a one-to-one function defined by $$f:{X} \rightarrow Y$$ with domain $$X$$ and range $$Y$$. Then, $$f^{-1}:{Y} \rightarrow X$$ is called the inverse function. Here, $$f^{-1}\left ( x\right )$$ does not mean $$\frac{1}{f\left (x \right )}$$.

A function $$f^{-1}$$ associates each element $$y$$ in $$Y$$ with one and only one element $$x$$ in $$X$$.

E2.9: Find inverse functions $$f^{-1}(x)$$.

An inverse function is a function that can reverse into another function. Now, let us understand the inverse function through a diagram.

In the above image, we will define a  function $$f$$ from the set $$X$$ to the set $$Y$$ in the form of $$f\left (x \right) =x-2$$. It means, when we subtract $$2$$ from all the elements of set $$X$$, it will give the elements of the set $$Y$$; where $$f$$ is the function, and $$x$$ is the input of the function.

Now, let us construct the inverse function $$ f^ {-1} $$. The relation in the inverse is represented by $$ f^ {-1} \left (x \right) =y+2$$. It means, when we add $$2$$  in all  the elements of $$Y$$, it gives all elements of $$X$$.

$$f\left (1 \right) =-1$$ that means $$f^ {-1} \left ( -1 \right) =1$$.

$$ f\left (3\right) =1$$ that mean $$ f^ {-1} \left (1\right) =3$$.

$$ f\left (4\right) =2$$ that mean $$ f^ {-1} \left (2\right) =4$$.

$$ f\left (7\right) =5$$ that mean $$f^ {-1} \left (5\right) =7$$.

In inverse function, the Domain of inverse function $$f^ {-1} $$ is the Range of the functions $$f$$, and the Range of the inverse functions $$f^ {-1} $$ is the Domain of the function $$f$$.

Worked examples of inverse functions

Example 1: Find the parameters $$a$$ and $$b$$ from the function $$f\left (x \right) =ax+b$$ so that $$ f^{-1}\left ( 2 \right )=4$$ and $$f^{-1}\left ( 6 \right )=2$$, where $$f^{-1}\left ( x \right )$$ is the inverse function of $$f$$.

Step 1: From the property of the inverse function.

$$ f^ {-1} \left (2 \right) =4$$, then $$f\left (4\right) =2$$, and $$ f^ {-1} \left (6\right) =2$$, then $$f\left (2\right) =6$$ 

Step 2: Use the above value, the equation becomes.

$$4a+b=2$$, and $$2a+b=6$$.

Step 3: Solve the above equation.

From the first equation, $$b=2-4a$$.

Step 4: Substitute the value of $$b$$ in the second equation.

$$2a+2-4a=6$$ gives $$a=-2$$.

Step 5: Solve the above equation.

$$a=-2$$.

Step 6: Substitute the value of $$a$$ in the first equation.

$$-8+b=2$$ gives $$b=10$$.

Step 7: Answer in preferred notation.

Therefore, the value of $$a=-2$$, and the value of $$b=10$$.

Example 2: If the function $$f$$ is defined as $$f\left (x \right) =\frac{5x-3}{4}$$, find $$f^ {-1}$$.

Step 1: Write the equation.

$$\frac{5x-3}{4} =y$$.

Step 2: Apply cross-multiplication.

$$5x-3=4y$$.

Step 4: Add $$3$$ on both sides of the equation.

$$5x=4y+3$$.

Step 5: Divide the equation $$55x=4y+3$$ by $$5$$ on both sides.

$$ x=\frac{4y+3}{5}$$

Step 6: Answer in preferred notation.

Therefore, $$f^ {-1} \left (y \right) =\frac{4y+3}{5} $$.

Example 3: If $$f\left (x \right) =x-4$$, evaluate $$ f^{-1} (-5)$$.

Step 1: From the given equation,

$$y=x-4$$

Step 2: The above equation can be written as:

$$x=y+4$$.

Therefore, $$f^ {-1} \left (y \right) =y+4$$.

Step 3: Substitute $$-5$$ to the equation.

$$f^{-1} (-5) =-5+4$$ gives $$f^{-1} ( -5) =-1$$.

Step 4: Answer in preferred notation.

Therefore, $$f^ {-1} (-5) =-1$$.