What is the "pre-image" of a function?
The "pre-image" of a function refers to the set of all input values that map to a specific output value. In other words, it is the collection of all elements in the domain of the function that are mapped to a particular element in the range.
To find the pre-image of a function for a given output value, you need to follow these steps:
1. Start with the output value for which you want to find the pre-image.
2. Look at the function definition, which describes how the input values are transformed into output values.
3. Reverse the process of the function by solving for the input variable.
4. Determine all the possible input values that would result in the desired output value.
5. Collect all these input values together to form the pre-image set.
For example, let's say you have a function f(x) = 2x, and you want to find the pre-image of the output value 10.
1. Start with the output value of 10.
2. Reverse the function by solving for x: 2x = 10. Divide both sides by 2 to get x = 5.
3. The input value 5 is the only possible value that maps to the output value 10.
4. Therefore, the pre-image of 10 under the function f(x) = 2x is {5}.
By following these steps, you can determine the pre-image of any output value under a given function.