A function relating set X to set Y is a special relation where each element x in X is mapped to one and only one element y in Y. (p.2)
Function notation: f : x → y or f(x) = y. x is the object and y (e.g. 2x + 1) is the image. (p.3)
A function is a one-to-one or a many-to-one relation. (p.3)
Vertical line test: if a vertical line cuts the graph at only one point, the relation is a function. If it cuts at no point or more than one point, it is not a function. (p.4)
Absolute value: |x| = x if x ≥ 0; |x| = −x if x < 0. The graph of f(x) = |x| is V-shaped with vertex (0, 0). (p.5)
Domain: the set of possible values of x. Codomain: the set of possible values of the function. Range: the actual values obtained from the domain (a subset of the codomain). (p.7)
Discrete function: points on the graph are separated and not connected. Continuous function: points are connected by a line or curve within the given interval. (p.7)
To sketch f(x) = |ax + b|, draw y = ax + b first, then reflect the part below the x-axis in the x-axis. (p.8)
Solving |ax + b| = k gives ax + b = −k or ax + b = k. For |ax + b| < k use −k < ax + b < k. (p.10)
1.2 Composite Functions
Composite function fg(x) = f[g(x)]: substitute g into f. gf(x) = g[f(x)]: substitute f into g. (p.13)
f2 = ff and g2 = gg. (p.14)
To find the image or object of a composite function, substitute the object into fg(x) or solve fg(x) = given image. (p.15)
If fg and f are given, find g by equating f[g(x)] with fg(x). If gf and f are given, let y = f(x), express x in terms of y, then replace y with x. (p.16)
Iterated functions f2, f3, f4 reveal a pattern (e.g. f(x) = 1/x2 gives fn(x) = x−2n for n odd and x2n for n even). (p.17)
1.3 Inverse Functions
f : x → y ⇔ f−1 : y → x. The graph of f−1 is the reflection of the graph of f in the line y = x. (p.20)
The notation f−1 is the inverse of f; f−1(x) ≠ 1/f(x). (p.21)
A function f has an inverse function f−1 if and only if f is a one-to-one function. (p.22)
f and g are inverses of each other if fg(x) = gf(x) = x. (p.23)
If f and g are inverses: domain of f = range of g, and domain of g = range of f. (p.23)
If the point (a, b) lies on the graph of f, then (b, a) lies on the graph of f−1. (p.24)
Horizontal line test: if a horizontal line cuts the graph at only one point, the function is one-to-one and has an inverse; if it cuts at two or more points, there is no inverse function. (p.24)
Finding f−1 algebraically: let y = f(x), make x the subject (x = f−1(y)), then replace y with x. (p.27)
Verify an inverse using ff−1(x) = f−1f(x) = x. (p.28)