
How do I take the inverse of an absolute value of function ?
Answer
472.5k+ views
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 .
We will find the inverse for the given absolute value function.
We know that the inverse for the given absolute value function is the same as the inverse for the function
Now, we will replace by , so we get
Now, we will replace by and by in the above equation, we get
Now, multiplying by a negative sign on both the sides of the equation, we get
We know that the product of two Negative Integers is always a Positive Integer. Thus, we get
Now, by solving for , we get
Thus, the inverse of the absolute value function, we get
Therefore, the inverse of the Absolute Value Function is .
Note:
We know that if a function which maps to , then the inverse of a function that maps to . 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.
Complete step by step solution:
We are given an absolute value function
We will find the inverse for the given absolute value function.
We know that the inverse for the given absolute value function
Now, we will replace
Now, we will replace
Now, multiplying by a negative sign on both the sides of the equation, we get
We know that the product of two Negative Integers is always a Positive Integer. Thus, we get
Now, by solving for
Thus, the inverse of the absolute value function, we get
Therefore, the inverse of the Absolute Value Function
Note:
We know that if a function
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