What is the Value of a Function at a Point?

S3-SA5-0192

Grade Level:

Class 10

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

Definition

What is it?

The 'value of a function at a point' tells us the output we get when we put a specific input into a function. Think of a function as a rule or a machine: you put something in, and it gives you something out. The value at a point is simply that 'something out' for a particular 'something in'.

Simple Example

Quick Example

Imagine a chai stall where the price of chai depends on how many spoons of sugar you ask for. Let's say the rule (function) is: Price = 10 + (2 * number of sugar spoons). If you ask for 3 spoons of sugar (your input), the value of the 'price function' at 3 spoons is 10 + (2 * 3) = 16 rupees. So, the value is 16.

Worked Example

Step-by-Step

Let's find the value of the function f(x) = 2x + 5 at the point x = 3.

1. Identify the function: f(x) = 2x + 5.
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2. Identify the point (input): x = 3.
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3. Substitute the value of x into the function: Replace every 'x' in the function with '3'. So, f(3) = 2 * (3) + 5.
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4. Perform the multiplication: f(3) = 6 + 5.
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5. Perform the addition: f(3) = 11.
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Answer: The value of the function f(x) = 2x + 5 at the point x = 3 is 11.

Why It Matters

Understanding function values is crucial for many advanced fields like AI/ML and Data Science, where you constantly evaluate models based on input data. Engineers use this to predict how structures will behave under different loads, and economists use it to forecast market trends. It's a fundamental skill for problem-solving in the real world.

Common Mistakes

MISTAKE: Multiplying the input value with the entire function, e.g., for f(x) = 2x + 5 at x=3, writing f(3) = 3 * (2x + 5) | CORRECTION: Only substitute the input value 'x' into the 'x' variable within the function, not the entire function expression.

MISTAKE: Forgetting the order of operations (BODMAS/PEMDAS) when calculating. E.g., for f(x) = 2x + 5 at x=3, calculating 2 + 3 * 5 = 25 instead of 2 * 3 + 5 = 11. | CORRECTION: Always follow BODMAS/PEMDAS rules: Brackets, Orders (powers/roots), Division/Multiplication (left to right), Addition/Subtraction (left to right).

MISTAKE: Confusing f(x) with 'f multiplied by x'. | CORRECTION: f(x) means 'the function f evaluated at the input x'. It's a notation for the output, not a multiplication.

Practice Questions

Try It Yourself

QUESTION: If a function is defined as g(y) = y^2 - 4, what is the value of g(5)? | ANSWER: g(5) = 5^2 - 4 = 25 - 4 = 21

QUESTION: Find the value of h(z) = (z + 1) / (z - 1) when z = 4. | ANSWER: h(4) = (4 + 1) / (4 - 1) = 5 / 3

QUESTION: A mobile data plan costs Rs. 100 for the first 1 GB and Rs. 15 for every extra GB. If C(g) represents the cost for 'g' GBs of data (where g > 1), and C(g) = 100 + 15 * (g - 1), what is the cost for 3 GB of data? | ANSWER: C(3) = 100 + 15 * (3 - 1) = 100 + 15 * 2 = 100 + 30 = Rs. 130

MCQ

Quick Quiz

If f(x) = x^2 + 3x - 2, what is f(2)?

The Correct Answer Is:

Substitute x=2 into the function: f(2) = 2^2 + 3*(2) - 2 = 4 + 6 - 2 = 10 - 2 = 8. So, option B is correct.

Real World Connection

In the Real World

When you book an auto-rickshaw using an app like Ola or Uber, the app calculates your fare based on the distance, time, and surge pricing. This fare calculation is a function, and when you enter your destination, the app finds the 'value of the fare function' for that specific trip. Similarly, ISRO scientists use function values to calculate rocket trajectories based on fuel consumption and thrust.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns exactly one output to each input | INPUT: The value you put into a function (often 'x') | OUTPUT: The value a function produces for a given input (often 'f(x)') | SUBSTITUTE: To replace a variable with a specific number or expression | EVALUATE: To calculate the numerical value of an expression or function at a specific point

What's Next

What to Learn Next

Now that you understand function values, you're ready to explore 'Graphing Functions'. Seeing how these input-output pairs look on a graph will give you a visual understanding and help you spot patterns and trends. Keep up the great work!