Chapter 3. Functions and Graphs

A function assigns to each input in the domain exactly one output in the range. Notation: f(x) [3][4].

Linear functions. y = mx + b has slope m = (y₂ − y₁)/(x₂ − x₁) and y-intercept b. Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Quadratic functions. y = ax² + bx + c graphs as a parabola with vertex at x = −b/(2a). It opens upward if a > 0 and downward if a < 0.

Transformations. For y = f(x): f(x) + k shifts up k units; f(x − h) shifts right h units; −f(x) reflects in the x-axis; af(x) stretches vertically by a factor a.

Composition and inverses. (f ∘ g)(x) = f(g(x)). The inverse f⁻¹ undoes f, so f⁻¹(f(x)) = x. A function has an inverse if it is one-to-one (passes the horizontal line test).

Other families: polynomial, rational, absolute value and radical functions. Key features to find are intercepts, asymptotes, maximum and minimum values, and end behaviour [4].

Review: If f(x) = 2x + 1, find f(3) and f⁻¹(x). (Answer: 7 and (x − 1)/2.)