Question

Let fand g be the functions from R to R such that$f(x)=2x$ and $g(x) = x ^ 2$ What is fg ?

• A. $$2x^3$$
• B. $$2x^2$$
• C. $$4x^5$$
• D. $$8x^3$$

Answer 1

Answer:

$$2x^3$$

Understanding Function Operations

• Functions can be combined using various algebraic operations to form new functions.

• For two functions f and g mapping from a set R to R (real numbers to real numbers), common operations include addition (f+g), subtraction (f-g), multiplication (fg or f*g), division (f/g), and composition (f o g).

Function Multiplication (fg)

• When the notation fg is used, it specifically represents the product of the functions f(x) and g(x).

• Mathematically, the product function is defined as (fg)(x) = f(x) * g(x). This means you multiply the output values of f and g for a given input x.

Step-by-Step Calculation

1. Given the function f(x) = 2x.

2. Given the function g(x) = x^2.

3. To find (fg)(x), we substitute the expressions for f(x) and g(x) into the definition of function multiplication: (fg)(x) = (2x) * (x^2).

4. Applying the rules of exponents (specifically, a^m * a^n = a^(m+n)), where x can be considered x^1: x * x^2 = x^(1+2) = x^3.

5. Therefore, (fg)(x) = 2 * x^3 = 2x^3.

Distinction: Function Multiplication vs. Function Composition

• It is crucial for competitive exams to understand the difference between function multiplication (fg(x) = f(x) * g(x)) and function composition ((f o g)(x) = f(g(x))).

• For example, with the given functions:

  • Function Multiplication: (fg)(x) = 2x * x^2 = 2x^3.

  • Function Composition: (f o g)(x) = f(g(x)) = f(x^2) = 2(x^2) = 2x^2.

  • Another composition: (g o f)(x) = g(f(x)) = g(2x) = (2x)^2 = 4x^2.

• These operations yield different results and notations, so accurate interpretation is vital.

Domain of Combined Functions

• For algebraic operations like multiplication, the domain of the resulting function (fg) is the intersection of the domains of the individual functions (f and g).

• Since both f(x) = 2x and g(x) = x^2 are defined for all real numbers (Domain = R), their product fg(x) = 2x^3 is also defined for all real numbers.

• Domain of f is R.

• Domain of g is R.

• Domain of fg is R ∩ R = R.