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Chapter 2

MA100 Chapter Notes - Chapter 2: Inverse Function

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MA100 Lab Notes - Transformations, Composition and Function Operations, Inverses
Text References: Precalc text - 1.5, 1.7, 5.1-5.2
Transformations and Translations
Consider the graph of the function f(x). Then, if c > 0,
f(x)±cis the graph of fshifted up [+] or down [] by cunits
f(x±c) is the graph of fshifted to the left [+] or to the right [] by cunits
cf (x) is the graph of fstretched [ if c > 1 ]
or compressed [ if c < 1 ] vertically by a factor of c
f(cx) is the graph of fcompressed [ if c > 1 )]
or stretched [ if c < 1 ] horizontally by a factor of c
f(x) is the graph of freflected in the x-axis
f(x) is the graph of freflected in the y-axis
You should know the shapes of basic curves such as y=x,y=x2,y=x3,y=x,y=1
xand y=|x|, and be
able to apply the above transformations to each.
Composition of functions:
Two functions can be combined to form a new function through a process called composition. The composition
of f(x) and g(x) is denoted (fg) (x) = f(g(x)). For example, if f(x) = x2+ 3xand g(x) = 2xthen
(fg) (x) = f(2x) = (2x)2+ 3 (2x) = 4x+ 6xwhile (gf) (x) = gx2+ 3x= 2x2+ 3x[ function
composition is not commutative, (fg) (x)6= (gf) (x) ]. Note that the domain of fgmust be contained
within the domain of g; i.e., DomfgDomg.
Inverse Functions:
If the function y=f(x) has an inverse function (i.e. f(x) is one-to-one), it is denoted by y=f1(x).[Note:
f1(x)6= [f(x)]1=1
f(x)(the reciprocal function)]
1. f(a) = bf1(b) = a
2. f(f1(x)) = f1(f(x)) = xfor all x
3. DomfRangef1and, similarily, Domf1Rangef
4. The curves f(x) and f1(x) are reflections in the line y=x
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