unit 3.4 & 3.5- august 21st

Vocabulary flashcards covering composition, domains, and transformations from the lecture notes.

Composition of functions (f ∘ g)

A composite function formed by applying g first, then f; (f ∘ g)(x) = f(g(x)).

Inner function

The inner part of a composition that is evaluated first, e.g., g(x) in f(g(x)).

Outer function

The function that is applied to the result of the inner function, e.g., f in f(g(x)).

Domain of a composition

The set of x values for which both g(x) is defined and f(g(x)) is defined.

Domain strategy for f ∘ g

Find the domain of g first, then ensure g(x) lies in the domain of f.

Decomposition of a function

Expressing a composite function as a composition of two simpler functions; there can be multiple valid decompositions.

Square root domain rule

For sqrt(u), you must have u ≥ 0; with nested roots, ensure inner expression makes the outer root defined.

Rational function domain rule

Exclude x-values that make a denominator zero (e.g., x cannot be 1 if the denominator is x−1).

Vertical shift

Moving a graph up or down by adding or subtracting a constant to the entire function: y = f(x) ± c.

Horizontal shift

Shifts left or right by the inside sign; plus inside means shift left, minus inside means shift right (opposite of sign).

Right shift example

f(x−3) shifts the graph to the right by 3.

Vertical reflection

A reflection across the x-axis produced by a negative sign in front of the function: y = −f(x).

Horizontal reflection

A reflection across the y-axis produced by a negative sign inside the function: f(−x).

Even function

A function with symmetry about the y-axis: f(−x) = f(x) for all x.

Odd function

A function with symmetry about the origin: f(−x) = −f(x) for all x.

Neither even nor odd

A function that is not symmetric as either even or odd.

Vertical stretch

Multiplying a function by a>1 stretches the graph; if 0<a<1, it compresses; negative a also reflects.

Absolute value transformation (basic)

The base function y = |x| is V-shaped and even; shifting/reflection follow the same inside/outside rules as other functions.

Decompose example (two-step decomposition)

Given a composite f ∘ g, choose an intermediate function h(x) so that f ∘ g = f ∘ h with h(x) = inside of the outer function; multiple valid decompositions may exist.

Inside function

The expression inside the outer function in a composition, e.g., g(x) in f(g(x)).

Outside function

The function applied after the inside function, i.e., the outer function in a composition.

Domain as a whole for f(g(x))

You must consider both the domain of g and the domain of f applied to g(x); some x are excluded because g(x) lands outside f’s domain.

Test for domain with fractions

When a fraction is involved, exclude x-values that make the denominator zero.

Test for domain with nested sqrt

For nested square roots, require inner expressions to be nonnegative; determine domain by ensuring radicands are ≥ 0.