What is the Image of an Element in a Function?

S3-SA5-0349

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

In a function, the 'image' of an element is the output value you get when you put that specific element into the function. It's like asking, 'What does this machine produce when I feed it this particular ingredient?'

Simple Example

Quick Example

Imagine a machine that takes a number and doubles it. If you put the number 5 into this machine, the output will be 10. Here, 10 is the image of 5 under this 'doubling' function.

Worked Example

Step-by-Step

Let's say we have a function f(x) = 2x + 3. We want to find the image of the element 4.

Step 1: Understand the function. The function f(x) = 2x + 3 means 'take any number x, multiply it by 2, and then add 3'.
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Step 2: Identify the element. The element we are interested in is 4.
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Step 3: Substitute the element into the function. Replace 'x' with '4' in the function's rule: f(4) = 2(4) + 3.
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Step 4: Perform the multiplication. f(4) = 8 + 3.
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Step 5: Perform the addition. f(4) = 11.
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Answer: The image of the element 4 under the function f(x) = 2x + 3 is 11.

Why It Matters

Understanding images is crucial because functions are everywhere, from predicting stock prices in economics to designing circuits in engineering. In AI/ML, functions help models learn patterns, and finding images allows us to see what output the model predicts for a given input, like how a face recognition system identifies a person from their photo.

Common Mistakes

MISTAKE: Confusing the element with its image, thinking they are the same value. | CORRECTION: Remember the element is the input (what goes into the function), and the image is the output (what comes out). They are usually different.

MISTAKE: Not following the order of operations (BODMAS/PEMDAS) when calculating the image. | CORRECTION: Always apply operations in the correct sequence (Brackets, Orders, Division/Multiplication, Addition/Subtraction) to avoid calculation errors.

MISTAKE: Applying the function's rule incorrectly, like adding before multiplying when it should be the other way around. | CORRECTION: Carefully read the function's definition and apply each operation exactly as specified.

Practice Questions

Try It Yourself

QUESTION: If a function g(y) = y - 7, what is the image of the element 15? | ANSWER: 8

QUESTION: For the function h(t) = t^2 + 1, find the image of the element -3. | ANSWER: 10

QUESTION: A mobile data plan charges Rs. 50 for the first 1 GB and Rs. 10 for every additional GB. If a function C(G) represents the cost for G GB (where G > 1), and C(G) = 50 + 10(G-1), what is the image of the element 3 GB? | ANSWER: Rs. 70

MCQ

Quick Quiz

For the function f(x) = 3x - 2, what is the image of the element 5?

The Correct Answer Is:

To find the image, substitute x=5 into the function: f(5) = 3(5) - 2 = 15 - 2 = 13. So, 13 is the correct image.

Real World Connection

In the Real World

Think about online food delivery apps like Swiggy or Zomato. If the delivery time is a function of distance, then for a specific restaurant (element) at a certain distance, the estimated delivery time (image) is calculated. This helps you decide if you want to order from that place.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns exactly one output to each input. | ELEMENT: An input value that goes into a function. | OUTPUT: The result produced by a function. | DOMAIN: The set of all possible input elements for a function. | RANGE: The set of all possible image values (outputs) of a function.

What's Next

What to Learn Next

Now that you understand what an image is, you can explore the concepts of 'Domain and Range of a Function'. These build directly on images by looking at all possible inputs and outputs a function can have.