What is f(g(x))?

S3-SA5-0048

Grade Level:

Class 9

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

Definition

What is it?

f(g(x)) is a 'function of a function', also called a composite function. It means you first put a value into the inner function, g(x), and then take that result and put it into the outer function, f(x). Think of it as a two-step process.

Simple Example

Quick Example

Imagine you have a machine that takes 'rupees' and gives you 'chai'. This is f(x). Now imagine another machine that takes 'coins' and converts them into 'rupees'. This is g(x). If you put 'coins' into the second machine, then take the 'rupees' it makes and put them into the first machine, you get 'chai'. This whole process is f(g(coins)).

Worked Example

Step-by-Step

Let's say f(x) = 2x + 1 and g(x) = x - 3. We want to find f(g(5)).

Step 1: First, find the value of the inner function, g(x), when x = 5.
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g(5) = 5 - 3
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g(5) = 2
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Step 2: Now, take this result (which is 2) and put it into the outer function, f(x). So we need to find f(2).
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f(2) = 2 * (2) + 1
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f(2) = 4 + 1
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f(2) = 5
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So, f(g(5)) = 5.

Why It Matters

Understanding f(g(x)) is crucial in many advanced fields. In AI/ML, it helps build complex models by combining simpler functions, like layers in a neural network. Engineers use it to model systems where one output becomes the input for another, helping design everything from bridges to rockets.

Common Mistakes

MISTAKE: Calculating g(f(x)) instead of f(g(x)). | CORRECTION: Always work from the inside out. First calculate the inner function (g(x)), then use that result in the outer function (f(x)).

MISTAKE: Multiplying f(x) and g(x) together. | CORRECTION: f(g(x)) is not multiplication. It's substitution – you replace 'x' in f(x) with the entire expression for g(x).

MISTAKE: Forgetting to substitute the entire g(x) expression into f(x) when finding f(g(x)) algebraically. | CORRECTION: If f(x) = 2x and g(x) = x+1, then f(g(x)) = 2(x+1), not 2x+1.

Practice Questions

Try It Yourself

QUESTION: If f(x) = x + 5 and g(x) = 2x, find f(g(3)). | ANSWER: 11

QUESTION: If f(x) = x^2 and g(x) = x - 1, find f(g(4)). | ANSWER: 9

QUESTION: If f(x) = 3x - 2 and g(x) = x^2 + 1, find f(g(x)). | ANSWER: 3x^2 + 1

MCQ

Quick Quiz

If f(x) = x + 10 and g(x) = x/2, what is f(g(4))?

The Correct Answer Is:

First, g(4) = 4/2 = 2. Then, f(g(4)) becomes f(2). Since f(x) = x + 10, f(2) = 2 + 10 = 12. So, option A is correct.

Real World Connection

In the Real World

Think about how your mobile phone processes a photo. First, the camera sensor (g(x)) captures light and converts it into raw digital data. Then, an image processing algorithm (f(x)) takes that raw data and enhances it, adjusts colours, and saves it as a JPEG. This entire process, from light to final image, is an example of f(g(x)).

Key Vocabulary

Key Terms

COMPOSITE FUNCTION: A function made by combining two or more functions, where the output of one becomes the input of another. | INNER FUNCTION: The function whose value is calculated first in a composite function (e.g., g(x) in f(g(x))). | OUTER FUNCTION: The function that uses the result of the inner function as its input (e.g., f(x) in f(g(x))). | SUBSTITUTION: Replacing a variable or expression with another equivalent value or expression.

What's Next

What to Learn Next

Great job understanding f(g(x))! Next, you can explore inverse functions, which 'undo' the effect of a function. This will help you understand how functions can be reversed, building on your knowledge of how they are combined.