Problem 94 - Algebra 2

Question

If \(f(x)=3 x-4\) and \(g(x)=x+3,\) what does \((f \cdot g)(x)\) mean? What is \((f \cdot g)(x) ?\) Simplify the answer.

Step-by-Step Solution

Answer

\((f \cdot g)(x) = 3x^2 + 5x -12\)

Step 1: Understand Notation

The notation \(f(x)=3x-4\) and \(g(x)=x+3\) represent two different functions. The variable \(x\) is the input value. The function \(f(x)\) multiplies the input by 3 and subtracts 4 whereas the function \(g(x)\) takes the input and adds 3.

Step 2: Define (f \cdot g)(x)

The notation \((f \cdot g)(x)\) represents the multiplication of functions \(f(x)\) and \(g(x)\). That is, \((f \cdot g)(x) = f(x) \cdot g(x)\).

Step 3: Substitute and Calculate

Substitute \(f(x)\) and \(g(x)\) into \((f \cdot g)(x)\) which results in \((3x-4)\cdot (x+3)\). Expand and simplify this expression and the final answer becomes \(3x^2 + 5x -12\).

Key Concepts

Understanding Function Notation

Function notation is a way to easily represent the relationship between input and output in mathematical functions. Here, the expression like \( f(x) = 3x - 4 \) is read as "\( f \) of \( x \)." It indicates a function named \( f \) where \( x \) is the input value.
- The output is calculated by following the specific rule provided by the function. In this instance, multiply \( x \) by 3 and subtract 4.
In general, function notation is:

• Easy to understand and interpret.
• Helps in organizing work systematically.
• Can represent complex expressions simply.

With function notation, tackling complex mathematical problems becomes more structured.

Introduction to Polynomial Functions

Polynomial functions involve expressions comprising variables, coefficients, and exponents. Here, both \( f(x) = 3x - 4 \) and \( g(x) = x + 3 \) are simple polynomial functions.
Polynomial functions can be categorized based on their degree:

• Linear polynomial: Degree is 1, like \( f(x) = 3x - 4 \).
• Quadratic polynomial: Degree is 2, like \( 3x^2 + 5x - 12 \) (from expanding \((f \cdot g)(x)\)).

Polynomial functions are:

• Easy to differentiate and integrate.
• Form the foundation for many algebraic computations.

Understanding how to manipulate these functions is a key skill in algebra and calculus.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form without changing their value. In the exercise, we simplified the expression \((3x-4)\cdot (x+3)\). Here's how you can do it:

• Use the distributive property (i.e., \( a(b + c) = ab + ac \)) to expand the terms.
• Apply this expansion: \( 3x(x+3) \) becomes \( 3x^2 + 9x \).
• Expand the second part: \(-4(x+3)\) becomes \(-4x - 12\).
• Combine all terms ensuring like terms are added together: \( 3x^2 + 9x - 4x - 12 = 3x^2 + 5x - 12 \).

Simplifying expressions makes calculation manageable and uncovering underlying properties easier for mathematical analysis.