Functions and Mathematical Models Free Online FlashCards

What is a function?

A function assigns exactly one output to each permitted input. The input is the independent variable, and the output is the dependent variable.

What does the vertical line test determine?

A graph represents a function if every vertical line intersects it at most once. This is the vertical line test.

How do domain and range differ?

The domain is the set of allowable inputs, while the range is the set of outputs produced by those inputs.

What is the domain of g(x) = (x + 1)/(x − 3)?

The domain is all real numbers except x = 3, so it is (−∞, 3) ∪ (3, ∞).

For g(x) = √(x + 4) − 1, what are the domain and range?

The domain is [−4, ∞), and the range is [−1, ∞). The graph is shifted left 4 units and down 1 unit from √x.

What do a, b, h, and k control in a function transformation?

In y = a f(b(x − h)) + k, a controls vertical stretch or reflection, b controls horizontal scaling or reflection, h shifts horizontally, and k shifts vertically.

In (f ∘ g)(x), which function acts first?

The composition is (f ∘ g)(x) = f(g(x)): g acts first, and f acts on that result.

Why is function composition generally not commutative?

Composition is generally not commutative: f ∘ g and g ∘ f can produce different functions because they apply the functions in different orders.

When does a function have an inverse function?

An inverse exists as a function only when the original function is one-to-one, meaning distinct inputs produce distinct outputs.

What does f⁻¹(x) mean?

f⁻¹(x) denotes the inverse function, not the reciprocal 1/f(x).

In P(t) = P₀bᵗ, what does b represent?

For P(t) = P₀bᵗ, b is the growth factor per time unit. A percentage growth rate r gives b = 1 + r.

How are logarithms and exponentials related?

A logarithm reverses exponentiation: log_b(x) = y exactly when bʸ = x, with b > 0 and b ≠ 1.