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Chapter 4: Problem 105

For each pair of functions, find \((f g)(x)\). $$ f(x)=2 x, \quad g(x)=5 x-1 $$

Short Answer

Expert verified
(fg)(x) = 10x^2 - 2x

Step by step solution

01

Understand the Problem

The task is to find the product of two functions, which is denoted by \( (fg)(x) \). We are given the functions \( f(x) = 2x \) and \( g(x) = 5x - 1 \).
02

Apply the Definition of Function Multiplication

To find \((fg)(x)\), multiply the given functions \(f(x)\) and \(g(x)\) together. This is written as: \[ (fg)(x) = f(x) \cdot g(x) \]
03

Perform the Multiplication

Substitute the expressions for \(f(x)\) and \(g(x)\): \[ (fg)(x) = (2x) \cdot (5x - 1) \]
04

Distribute and Simplify

Use the distributive property of algebra to expand the product: \[ 2x \cdot 5x + 2x \cdot (-1) = 10x^2 - 2x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Algebra Basics
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It helps in expressing general mathematical relationships and understanding the structure behind various calculations.
Here鈥檚 why algebra is important:
  • It simplifies complex problems into manageable parts.
  • It allows us to formulate problems so that we can solve them step-by-step.

In the given problem, the functions are defined as algebraic expressions:
$$ f(x) = 2x $$ and $$ g(x) = 5x - 1 $$.
To solve the problem, we need to understand how to manipulate and combine these algebraic expressions.
Using the Distributive Property
The distributive property is a fundamental property of algebra used to multiply a single term by two or more terms inside a parenthesis.
It states that for any numbers or expressions, the following equation holds:
$$ a(b + c) = ab + ac $$
In our problem, we need to use the distributive property to multiply the functions:
$$(fg)(x) = (2x) \times (5x - 1)$$.
Applying the distributive property, we first multiply \(2x\) by \(5x\) and then by \(-1\):
$$ 2x \times 5x + 2x \times (-1) = 10x^2 - 2x $$
This property helps to break down the multiplication into smaller, more manageable parts.
Simplifying Expressions
Simplifying an algebraic expression means reducing it to its simplest form. This involves combining like terms and performing arithmetic operations.
In our example, after using the distributive property, we get:
$$ 10x^2 - 2x $$
This final expression is simplified as it cannot be reduced further. Each term is distinct:
  • $$10x^2$$ represents the quadratic term.
  • $$-2x$$ represents the linear term.

To simplify expressions effectively:
  • Ensure all like terms are combined.
  • Double-check multiplication and addition steps.
  • Factor common terms if necessary.

Simplifying makes it easier to interpret the results and apply them to real-world problems.

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