Algebra of Functions and Function Composition

What is the simplified form of the difference quotient for the function f(x) = 4x + 7?

Which of the following represents the value of f(3) for the function f(x) = 2x² – 3x + 4?

What operation does (f * g)(-2) = (f(-2))(g(-2)) represent?

When calculating the difference quotient for f(x) = 2x² – 3x + 4, what is the final simplified form?

What is the correct expression for the sum of the functions f(x) = x² + 1 and g(x) = 3x – 5?

What is the domain for the quotient of the functions m(x) and n(x)?

What is the simplified expression for the product of f(x) = x² – 5 and g(x) = √(6 – x)?

What is the result of (f - g)(x) for f(x) = x² + 1 and g(x) = 3x - 5?

Which of the following represents the product (fg)(4) for f(x) = x² + 1 and g(x) = 3x - 5?

How is the domain of the quotient function (f / g)(x) determined?

How would you describe the domain of the function g(x) = √(6 - x)?

What is the expression for (f / g)(x) when f(x) = x² + 1 and g(x) = 3x - 5?

Algebra of Functions and Function Composition

The sum, difference, product, and quotient of functions are defined for all x common to the domains of both functions.
Sum: (f + g)(x) = f(x) + g(x)
Difference: (f – g)(x) = f(x) – g(x)
Product: (fg)(x) = (f(x))(g(x))
Quotient: (f / g)(x) = f(x) / g(x), provided g(x) ≠ 0

Examples of Function Operations

Given f(x) = x² + 1 and g(x) = 3x – 5:
- (f + g)(x) = x² + 3x – 4, domain: (-∞, ∞)
- (f – g)(x) = x² – 3x + 6, domain: (-∞, ∞)
- (fg)(x) = 3x³ – 5x² + 3x – 5, domain: (-∞, ∞)
- (f / g)(x) = (x² + 1) / (3x – 5), domain: (-∞, 5/3) U (5/3, ∞)
- (f + g)(3) = 14
- (fg)(4) = 119
Given f(x) = x² – 5 and g(x) = √(6 – x)
- (fg)(x) = x²√(6 – x) – 5√(6 – x), domain: (-∞, 6]
Given m(x) = (x – 4) / (x² – 25) and n(x) = (x – 4) / (5 – x)
- (m / n)(x) = -1 / (x + 5), x ≠ -5, 5, 4, domain: (-∞, -5) U (-5, 4) U (4, 5) U (5, ∞)
- (m + n)(x) = (x³ – x² – 16x – 20) / ((x² – 25)(5 – x)), domain: (-∞, -5) U (-5, 5) U (5, ∞)

Difference Quotient

The difference quotient for a function f(x) is given by:
- ( f(x + h) – f(x) ) / h
This expression determines the average rate of change of the function f(x) over an interval of length h.

Examples of Difference Quotient

Given f(x) = 4x + 7:
- f(x + h) – f(x) / h = 4, h ≠ 0
Given f(x) = 2x² – 3x + 4:
- f(x + h) – f(x) / h = 4x + 2h – 3, h ≠ 0

Test your understanding of function operations such as sum, difference, product, and quotient of functions. This quiz covers various examples and their corresponding domains, helping you master the composition of functions and their algebraic properties.