LIMITS AND CONTINUITY OF FUNCTIONS

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1. FUNCTIONS

HISTORY

The idea of a function goes back to Sharaf al-Dn al-Ts, a Persian mathematician in the 12th century.

However, it was Gottfried Leibniz who coined term function in a 1673 letter to describe a quantity related

to a curve. The function described by Leibniz is now known as the differentiable function. This function

includes the concepts of limits and derivatives, which are the basis of calculus. During the mid-18th

century, the word function was used by Leonhard Euler to describe an expression or formula involving

various arguments.

DOMAIN AND RANGE OF A FUNCTION

The domain of the function f, denoted by dom f or f

D , is the whole set of permissible values of x. While

its range, denoted by ran f or f

R , is the set of corresponding elements of y.

COMMON TYPES OF FUNCTIONS

The following is a list of common functions, describing their forms, graph, domain, and range.

1. Constant Function

Form: ( ) cxf = , where c Graph: Horizontal line with y-intercept c

Domain: Set of real numbers, denoted as { }xx Range: { }cyy =

2. Linear Function

Form: ( ) bmxxf += , where 0,, mbm Graph: Line with slope m and y-intercept b

Domain: { }xx Range: { }yy

Definition: A function f from a set X to a set Y is a rule of correspondence that pairs to

each element x of X exactly one element y of Y. This correspondence can be

denoted by

yxf : or ( ) yxf = .

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3. Quadratic Function

Form: ( ) cbxaxxf ++= 2 , where 0,,, acba Graph: Vertical parabola with the following properties

concave upward if 0>a , or concave downward if 0

LIMITS AND CONTINUITY OF FUNCTIONS

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7. Piecewise Function

Form: ( )( )( )

( )

=

nn Ixxg

Ixxg

Ixxg

xf

,

,

,

22

11

MM, where

nIII ,,, 21 K are disjoint intervals of x

Graph: Vary from piecewise function to piecewise function

Domain: The union of the nIII ,,, 21 K intervals, except the numbers at which any of the

functions ( ) ( ) ( )xgxgxgn

,,,21

K is undefined with respect to i

I , where ni ,,2,1 K=

Range: Vary from piecewise function to piecewise function

MISCELLANEOUS FUNCTIONS

1. Absolute Value Function

Form: ( ) ( ) ( ) ( )( ) ( )

a or decreasing if 10 yy

5. Natural Logarithmic Funciton and Other Logarithmic Functions

Form: ( ) ( )xxf ln= ( ) xxf

alog=

Graph: Has the following characteristics

has a vertical asymptote at 0=x

x-intercept is 1

increasing if 1>a or decreasing if 10 xx Range: { }yy

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EXERCISES

Determine the domain and range of the following functions. Sketch their graphs.

1. ( ) 42 = xxf 6. ( )3

322

=

x

xxxp

2. 228 xxy = 7. ( )2

43

=

x

xxxr

3. ( )32+= xy 8. ( )1

2

2

=

x

xxh

4. x

y1

= 9. 2

4 xy =

5. ( )2

1

+

=

x

xxg 10. ( ) 92 = xxq

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11. ( ) 3 1+= xxf 16. ( ) 29 xxf =

12. ( ) 42 += xxg 17. 13 += xy

13. ( )1

=

x

xxh 18. 2+= xy

14. ( )

>

=