Chapter 1 | Functions and Graphs
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• Point-slope equation of a line y − y 1 = m ( x − x 1 ) • Slope-intercept form of a line y = mx + b • Standard form of a line ax + by = c • Polynomial function f ( x ) = a n x n + a n −1 x • Generalized sine function f ( x ) = A sin ⎛ ⎝ B ( x − α ) ⎞ ⎠ + C • Inverse functions f −1 ⎛ ⎝ f ( x ) ⎞
n −1 +⋯+ a
1 x + a 0
⎠ = x for all x in D , and f ⎛
⎞ ⎠ = y for all y in R .
⎝ f −1 ( y )
KEY CONCEPTS 1.1 Review of Functions • A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input. • If no domain is stated for a function y = f ( x ), the domain is considered to be the set of all real numbers x for which the function is defined. • When sketching the graph of a function f , each vertical line may intersect the graph, at most, once. • A function may have any number of zeros, but it has, at most, one y -intercept. • To define the composition g ∘ f , the range of f must be contained in the domain of g . • Even functions are symmetric about the y -axis whereas odd functions are symmetric about the origin. 1.2 Basic Classes of Functions • The power function f ( x ) = x n is an even function if n is even and n ≠0, and it is an odd function if n is odd. • The root function f ( x ) = x 1/ n has the domain [0, ∞) if n is even and the domain (−∞, ∞) if n is odd. If n is odd, then f ( x ) = x 1/ n is an odd function. • The domain of the rational function f ( x ) = p ( x )/ q ( x ), where p ( x ) and q ( x ) are polynomial functions, is the set of x such that q ( x ) ≠0. • Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions. • A polynomial function f with degree n ≥1 satisfies f ( x )→±∞ as x →±∞. The sign of the output as x →∞ depends on the sign of the leading coefficient only and on whether n is even or odd. • Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the x - and y -axes are examples of transformations of functions. 1.3 Trigonometric Functions • Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180° has a radian measure of π rad.
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