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What is the Pre-image of an Element?
Grade Level:
Class 10
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The pre-image of an element is the input value (or values) that, when put into a function, gives you that specific output value. Think of it as finding what 'started' a particular result. It's like tracing back from the answer to the question.
Simple Example
Quick Example
Imagine a function where 'adding 5 to your current age' gives your future age. If your future age (the output) is 17, then the pre-image is your current age, which would be 12. You trace back from 17 to find 12.
Worked Example
Step-by-Step
Let's say a mobile data plan charges you Rs 20 for every GB used, plus a fixed monthly charge of Rs 50. The function is: Cost = 20 * (GB used) + 50.
---Suppose your bill (Cost) for a month was Rs 150. What is the pre-image, meaning how many GB did you use?
---Step 1: Write down the given function and the output: 150 = 20 * (GB used) + 50.
---Step 2: Isolate the term with 'GB used'. Subtract 50 from both sides: 150 - 50 = 20 * (GB used).
---Step 3: Simplify: 100 = 20 * (GB used).
---Step 4: Divide both sides by 20 to find 'GB used': 100 / 20 = GB used.
---Step 5: Calculate the result: 5 = GB used.
---Answer: The pre-image for a bill of Rs 150 is 5 GB. You used 5 GB of data.
Why It Matters
Common Mistakes
MISTAKE: Confusing the pre-image with the image (output). Students might be given the pre-image and asked for the pre-image. | CORRECTION: The pre-image is always the input value(s). The image is the output value(s). Always identify which one you are given and which you need to find.
MISTAKE: Not performing inverse operations correctly when solving for the pre-image. Forgetting to subtract when adding, or divide when multiplying. | CORRECTION: To find the pre-image, you must 'undo' the function. This means performing the opposite operation in reverse order of operations (PEMDAS/BODMAS).
MISTAKE: Assuming there is always only one pre-image for every image. | CORRECTION: For some functions (like x^2), an output can have multiple pre-images (e.g., if output is 9, pre-images are 3 and -3). Always consider if there could be more than one input that gives the same output.
Practice Questions
Try It Yourself
QUESTION: A function doubles a number and then adds 3. If the output (image) is 15, what is the pre-image? | ANSWER: 6
QUESTION: A function f(x) = 3x - 7. If the image is 20, find the pre-image. | ANSWER: 9
QUESTION: The function for converting temperature from Celsius (C) to Fahrenheit (F) is F = (9/5)C + 32. If a dish needs to be cooked at 212 degrees Fahrenheit, what is the pre-image in Celsius? | ANSWER: 100 degrees Celsius
MCQ
Quick Quiz
For the function f(x) = x^2 + 1, what is a pre-image of the element 10?
10
3
9
101
The Correct Answer Is:
B
If f(x) = 10, then x^2 + 1 = 10. Subtracting 1 gives x^2 = 9. Taking the square root gives x = 3 or x = -3. So, 3 is a pre-image of 10. Options A, C, and D are incorrect as they don't satisfy the equation.
Real World Connection
In the Real World
When you use a food delivery app like Swiggy or Zomato, the app calculates the total cost based on food price, delivery fee, and discounts. If you see a total bill of Rs 300, and you want to know what was the original food price before discounts and delivery, you are essentially finding the pre-image. This 'reverse calculation' helps you understand the components of the final amount.
Key Vocabulary
Key Terms
FUNCTION: A rule that assigns each input exactly one output | IMAGE: The output value of a function for a given input | INPUT: The value put into a function | OUTPUT: The value produced by a function | INVERSE OPERATION: An operation that undoes another operation (e.g., addition undoes subtraction)
What's Next
What to Learn Next
Now that you understand pre-images, you're ready to explore 'Inverse Functions'. Inverse functions are special types of functions that directly help you find the pre-image for any output, making these calculations even faster and more systematic!
