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**preview**shows half of the first page. to view the full**1 pages of the document.**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 (f◦g) (x) = f(g(x)). For example, if f(x) = x2+ 3xand g(x) = 2√xthen

(f◦g) (x) = f(2√x) = (2√x)2+ 3 (2√x) = 4x+ 6√xwhile (g◦f) (x) = gx2+ 3x= 2√x2+ 3x[ function

composition is not commutative, (f◦g) (x)6= (g◦f) (x) ]. Note that the domain of f◦gmust be contained

within the domain of g; i.e., Domf◦g⊆Domg.

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=f−1(x).[Note:

f−1(x)6= [f(x)]−1=1

f(x)(the reciprocal function)]

•Properties:

1. f(a) = b⇔f−1(b) = a

2. f(f−1(x)) = f−1(f(x)) = xfor all x

3. Domf≡Rangef−1and, similarily, Domf−1≡Rangef

4. The curves f(x) and f−1(x) are reﬂections in the line y=x

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