Calculus I > Precalculus Review
Review Objective
Basic Concepts of Functions
Defining a Function
Intervals
Implied Domain
Even and Odd Functions
Composite functions
Inverse Functions
Polynomial Functions
Linear Functions
Quadratic Functions
Polynomial Function
Rational Functions
Rational Functions
Vertical and Horizontal Asymptote
Exponential and Logarithm
Rational and Irrational Exponents
Exponential Functions
Logarithmic Functions
Laws of Logarithms
Trigonometric functions
Radian and right triangles
Trigonometric functions
Trigonometric functions on a coordinate system
Sine and cosine function
Tangent function
Trigonometric additions and subtractions
Trigonometric formulas
Inverse sine function
Inverse cosine function
Inverse tangent function

Composite functions

Given two functions $ f$ and $ g$, the composite function $ f\circ g$ is defined by

$\displaystyle (f\circ g)(x) = f(g(x)). $

Graph of a composite function. If we know the graph of  $ f(x) = x^2$, we can sketch the graphs of $ f(x) + 4 = x^2 + 4$, $ f(x-4) = (x - 4)^2$, etc.

Equation of Effect on the graph of $ f$
$ {y = f(x) + c}$ Shift it vertically upward if $ c > 0$, or downward if $ c < 0$.
$ {y = f(x + c)}$ Shift it horizontally to the left if $ c > 0$, or to the right if $ c < 0$.
$ {y = c f(x)}$ Stretch it vertically if $ 1 < c$, or compress it vertically if $ 0 < c < 1$.
$ {y = f(c x)}$ Compress it horizontally if $ 1 < c$, or stretch it horizontally if $ 0 < c < 1$.

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