Algebra Class 10: Combinations of Functions

Domains of Functions

• Real numbers causing division by zero are excluded from a function's domain.
• Square roots of negative numbers are not included in the domain of a function.
• The domain of f(x) = 9x + 5 is all real numbers, expressed as (-infinity, infinity).
• For g(x) = 3 / (x + 7), exclude -7 from its domain: (-infinity, -7) U (-7, infinity).
• The domain of h(x) = 1/x + 3/(x + 1) excludes 0 and -1: (-infinity, -1) U (-1, 0) U (0, infinity).

Composite Functions

• Composition of functions f and g, denoted as fog, is defined by (fog)(x) = f(g(x)).
• To find fog, replace g(x) in the function f with g(x).
• Composition of functions g and f, denoted as gof, is defined by (gof)(x) = g(f(x)).
• fog and gof represent different functions; they are not the same.

Function Composition Truths

• The equation f(g(x)) does not equal f(x) * g(x).
• For f(g(x)) = 3/g(x) - 9 with g(x) = 27/x, exclude both 0 and 3 from the domain of fog.

Finding Domains of Specific Functions

• The function f(x) = 14(x - 10) has a domain of (-infinity, infinity).
• The function f(x) = 12/x + 8 restricts its domain to (-infinity, -8) U (-8, infinity).
• For f(x) = 48/(x^2 + 12x - 63), identify roots through factorization resulting in domain (-infinity, -9) U (-9, 7) U (7, infinity).
• In f(x) = 5/(x^2 + 1) + 3/(x^2 - 16), only x^2 - 16 causes division by zero, leading to domain (-infinity, -4) U (-4, 4) U (4, infinity).

Square Root Functions

• For functions involving square roots, only non-negative radicands are allowed.
• For f(x) = √(x - 9), the domain is [9, infinity) since x - 9 must be greater than or equal to zero.
• For f(x) = √(x - 4) + √(x + 13), impose conditions on both radicands: x - 4 ≥ 0 and x + 13 ≥ 0. This yields a domain of [4, infinity).

Key Terms

• Radicand refers to the number or expression inside a radical symbol.