Question

How do I take the inverse of an absolute value of function \[f\left( x \right) = \left| {x - 2} \right|\]?

Answer 1

Hint: Here, we will use the condition for the absolute value of the function. Then by swapping the variables and solving for the variable, we will find the inverse of the given absolute function. The absolute value is defined as the non-negative integer without regard to its sign, it can be either a positive or negative integer.

Complete step by step solution:
We are given an absolute value function \[f\left( x \right) = \left| {x - 2} \right|\].
We will find the inverse for the given absolute value function.
We know that the inverse for the given absolute value function \[f\left( x \right) = \left| {x - 2} \right|\]is the same as the inverse for the function \[f\left( x \right) = - \left( {x - 2} \right)\]
\[ \Rightarrow f\left( x \right) = - \left( {x - 2} \right)\]
Now, we will replace \[f\left( x \right)\] by \[y\], so we get
\[ \Rightarrow y = - \left( {x - 2} \right)\]
Now, we will replace \[x\] by \[y\] and \[y\] by \[x\] in the above equation, we get
\[ \Rightarrow x = - \left( {y - 2} \right)\]
Now, multiplying by a negative sign on both the sides of the equation, we get
\[ \Rightarrow - x = \left( - \right)\left( - \right)\left( {y - 2} \right)\]
We know that the product of two Negative Integers is always a Positive Integer. Thus, we get
\[ \Rightarrow - x = y - 2\]
Now, by solving for \[y\], we get
\[ \Rightarrow y = - x + 2\]
Thus, the inverse of the absolute value function, we get
\[ \Rightarrow {f^{ - 1}}\left( x \right) = - x + 2\]

Therefore, the inverse of the Absolute Value Function \[f\left( x \right) = \left| {x - 2} \right|\] is \[{f^{ - 1}}\left( x \right) = - x + 2\].

Note:
We know that if a function\[f\] which maps \[x\] to\[y\], then the inverse of a function \[{f^{ - 1}}\] that maps \[y\] to \[x\]. If the horizontal line intersects the graph of the function in all places at exactly one point, then the given function should have an inverse that is also a function. If at any point, there are two or more points on the graph, in any horizontal line, it has no inverse. This test is known as the Horizontal Line Test to check whether a function has an inverse.