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Operations on Functions

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Operations on Functions

Study Guide

Key Definition

For functions $f(x)$ and $g(x)$, operations are defined as: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x) \times g(x)$, $(f/g)(x) = \frac{f(x)}{g(x)}$ where $g(x) \neq 0$.

Important Notes

Operations are only valid within the overlapping domain of $f(x)$ and $g(x)$.
The composition of functions $(f \circ g)(x)$ is defined as $f(g(x))$.
Ensure $g(x) \neq 0$ when performing division.
Check domain restrictions for each function individually.
Function operations can change the domain of the resulting function.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Operations on functions allow us to combine multiple functions to form new ones, providing flexibility in modeling real-world phenomena using $f(x)$ and $g(x)$.

Remember

Always verify the domain of your resulting function after performing operations.

Quick Reference

Sum of Functions:$(f+g)(x) = f(x) + g(x)$

Difference of Functions:$(f-g)(x) = f(x) - g(x)$

Product of Functions:$(fg)(x) = f(x) \times g(x)$

Quotient of Functions:$(f/g)(x) = \frac{f(x)}{g(x)}$

Composition of Functions:$(f \circ g)(x) = f(g(x))$

Understanding Operations on Functions

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Adding two functions means adding their outputs for the same input. For example, $(f+g)(x)=f(x)+g(x)$ means at each x you compute f(x) and g(x), then add those values.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $f(x) = 2x + 1$ and $g(x) = x^2 - 4$, what is $(f+g)(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A student wants to combine two savings plans represented by $f(x) = 5x + 10$ and $g(x) = 3x^2$. Find the total savings function.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given $h(x) = x^3 - 2x$, find the domain of $(h/g)(x)$ where $g(x) = x^2 - 4$.

Challenge Quiz

Single Choice Quiz

Advanced

Given $f(x) = x^2 + 3x$ and $g(x) = x - 5$, find the composition $(f \circ g)(x)$.

Recap

Watch & Learn

Review key concepts and takeaways