Combinations and Compositions of Functions: Precalculus Study Notes

Combinations and Compositions of Functions

Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0. There are several methods to solve them, including factoring and the square root property.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Square Root Property: If the equation is in the form x2 = k, then x = \pm\sqrt{k}.

Example: Solve x^2 - 6x + 8 = 0 by factoring.

• Factor: (x - 4)(x - 2) = 0

• Solutions: x = 4 and x = 2

Example: Solve x^2 = 16 by the square root property.

• Take square roots: x = \pm 4

Finding a Function’s Domain from its Equation

The domain of a function is the set of all real numbers for which the function is defined. Exclude values that cause division by zero or even roots of negative numbers.

• Division by zero: Denominators cannot be zero.

• Even roots: The radicand (expression under the root) must be non-negative for even roots.

Example: Find the domain of g(x) = \frac{5x}{x^2 - 49}.

• Set denominator not equal to zero: x^2 - 49 \neq 0 \implies x \neq \pm 7

• Domain: All real numbers except x = 7 and x = -7

The Algebra of Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the intersection of the domains of the original functions (and for division, exclude where the denominator is zero).

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\ g(x) \neq 0$

Example: Let f(x) = x - 5 and g(x) = x^2 - 1. Find (f + g)(x) and its domain.

• (f + g)(x) = (x - 5) + (x^2 - 1) = x^2 + x - 6

• Domain: All real numbers

Application: Modeling Population Change

Functions can model real-world data, such as birth and death rates. The change in population can be modeled as the difference between the birth and death functions.

• Births: $B(x) = -2.6x^2 + 33x + 4043$

• Deaths: $D(x) = 2x^2 - 13x + 2428$

• Population Change Function: $P(x) = B(x) - D(x)$

Example: Find the change in U.S. population in 2014 (where x = 14).

• Calculate: $P(14) = B(14) - D(14)$

• Substitute values and compute to find the result.

Additional info: The bar graph visually compares the number of births and deaths in the U.S. from 2000 to 2016, supporting the population change modeling example above.

The Composition of Functions

The composition of two functions f and g, denoted f ∘ g, is defined as (f ∘ g)(x) = f(g(x)). This means you apply g to x first, then apply f to the result.

• Order matters: f ∘ g is generally not the same as g ∘ f.

• Domain: The domain of f ∘ g consists of all x in the domain of g such that g(x) is in the domain of f.

Example: Given f(x) = 2x - 1 and g(x) = x^2 + 3, evaluate (f ∘ g)(4).

• First, find g(4) = 4^2 + 3 = 19

• Then, f(19) = 2 \times 19 - 1 = 37

• So, (f ∘ g)(4) = 37

Domain of Composition of Functions

To find the domain of a composite function f ∘ g:

1. Find the domain of g.

2. Find the domain of f.

3. The domain of f ∘ g is all x in the domain of g such that g(x) is in the domain of f.

Example: Given f(x) = 1 - x and g(x) = x + 5, find (f ∘ g)(x) and its domain.

• (f ∘ g)(x) = f(g(x)) = 1 - (x + 5) = -x - 4

• Domain: All real numbers

Decomposing Functions

Decomposing a function means expressing it as a composition of two simpler functions. If h(x) = f(g(x)), then f and g are the component functions.

• Example: Express h(x) = x^2 + 5 as a composition.

• Let g(x) = x and f(x) = x^2 + 5, so h(x) = f(g(x)).

• Alternatively, let g(x) = x^2 and f(x) = x + 5, so h(x) = f(g(x)) = g(x) + 5.

Additional info: Decomposition is useful for understanding the structure of complex functions and for simplifying operations such as differentiation and integration in calculus.