Function $$f ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } + x - 12 } { x - 3 } & ( x \neq 3 ) \\ a & ( x = 3 ) \end{array} \right.$$ When this function is continuous for all real numbers $x$, what is the value of $a$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

For the function $f ( x ) = x ^ { 2 } - 4 x + a$ and the function $g ( x ) = \lim _ { n \rightarrow \infty } \frac { 2 | x - b | ^ { n } + 1 } { | x - b | ^ { n } + 1 }$, let $h ( x ) = f ( x ) g ( x )$. What is the value of $a + b$, the sum of the two constants $a , b$ such that the function $h ( x )$ is continuous for all real numbers $x$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

The graph of a function $y = f ( x )$ defined on all real numbers is as shown in the figure, and a cubic function $g ( x )$ has leading coefficient 1 and $g ( 0 ) = 3$. When the composite function $( g \circ f ) ( x )$ is continuous on all real numbers, what is the value of $g ( 3 )$? [4 points]
(1) 31
(2) 30
(3) 29
(4) 28
(5) 27

For the function
$$f ( x ) = \begin{cases} x + 1 & ( x \leq 0 ) \\ - \frac { 1 } { 2 } x + 7 & ( x > 0 ) \end{cases}$$
Find the sum of all real values of $a$ such that the function $f ( x ) f ( x - a )$ is continuous at $x = a$. [4 points]

For the function $$f ( x ) = \begin{cases} 2 x + 10 & ( x < 1 ) \\ x + a & ( x \geq 1 ) \end{cases}$$ find the value of the constant $a$ such that $f$ is continuous on the entire set of real numbers. [3 points]

Two functions $$f ( x ) = \left\{ \begin{array} { l l } x + 3 & ( x \leq a ) \\ x ^ { 2 } - x & ( x > a ) \end{array} , \quad g ( x ) = x - ( 2 a + 7 ) \right.$$ Find the product of all real values of $a$ such that the function $f ( x ) g ( x )$ is continuous on the entire set of real numbers. [4 points]

For two functions $$\begin{aligned} & f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } - 4 x + 6 & ( x < 2 ) \\ 1 & ( x \geq 2 ) \end{array} , \right. \\ & g ( x ) = a x + 1 \end{aligned}$$ When the function $\frac { g ( x ) } { f ( x ) }$ is continuous on the entire set of real numbers, what is the value of the constant $a$? [4 points]
(1) $- \frac { 5 } { 4 }$
(2) $- 1$
(3) $- \frac { 3 } { 4 }$
(4) $- \frac { 1 } { 2 }$
(5) $- \frac { 1 } { 4 }$

Consider the function $$f(x) = \begin{cases} 3x - a & (x < 2) \\ x^2 + a & (x \geq 2) \end{cases}$$ If $f$ is continuous on the set of all real numbers, find the value of the constant $a$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The function $$f ( x ) = \begin{cases} 3 x - 2 & ( x < 1 ) \\ x ^ { 2 } - 3 x + a & ( x \geq 1 ) \end{cases}$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $a , b \in R , ( a \neq 0 )$. If the function $f$, defined as $$f ( x ) = \left\{ \begin{array} { c } \frac { 2 x ^ { 2 } } { a } , 0 \leq x < 1 \\ a , 1 \leq x < \sqrt { 2 } \\ \frac { 2 b ^ { 2 } - 4 b } { x ^ { 3 } } , \sqrt { 2 } \leq x < 8 \end{array} \right.$$ is continuous in the interval $[ 0 , \infty )$, then an ordered pair $( a , b )$ can be
(1) $( - \sqrt { 2 } , 1 - \sqrt { 3 } )$
(2) $( \sqrt { 2 } , - 1 + \sqrt { 3 } )$
(3) $( \sqrt { 2 } , 1 - \sqrt { 3 } )$
(4) $( - \sqrt { 2 } , 1 + \sqrt { 3 } )$

If the function $f ( x ) = \left\{ \begin{array} { l } a | \pi - x | + 1 , x \leq 5 \\ b | x - \pi | + 3 , x > 5 \end{array} \right.$ is continuous at $x = 5$, then the value of $a - b$ is:
(1) $\frac { 2 } { 5 - \pi }$
(2) $\frac { - 2 } { \pi + 5 }$
(3) $\frac { 2 } { \pi + 5 }$
(4) $\frac { 2 } { \pi - 5 }$

$f ( x ) = \left\{ \begin{array} { c l } \frac { \sin ( a + 2 ) x + \sin x } { x } & ; x < 0 \\ b & ; x = 0 \\ \frac { \left( x + 3 x ^ { 2 } \right) ^ { 1 / 3 } - x ^ { 1 / 3 } } { x ^ { 1 / 3 } } & ; x > 0 \end{array} \right.$ is continuous at $x = 0$, then $a + 2 b$ is equal to:
(1) 1
(2) - 1
(3) 0
(4) - 2

Consider the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { c l } \frac { \mathrm { a } \left( 7 \mathrm { x } - 12 - \mathrm { x } ^ { 2 } \right) } { \mathrm { b } \left| \mathrm { x } ^ { 2 } - 7 \mathrm { x } + 12 \right| } & , \mathrm { x } < 3 \\ 2 ^ { \frac { \sin ( \mathrm { x } - 3 ) } { \mathrm { x } - [ \mathrm { x } ] } } & , \mathrm { x } > 3 \\ \mathrm {~b} & , \mathrm { x } = 3 \end{array} \right.$, where $[ \mathrm { x } ]$ denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs $( \mathrm { a } , \mathrm { b } )$ such that $\mathrm { f } ( \mathrm { x } )$ is continuous at $x = 3$, then the number of elements in S is:
(1) 2
(2) Infinitely many
(3) 4
(4) 1

For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to : (1) 6 (2) 4 (3) 5 (4) 8

Below is the graph of the function f. If the function $( \mathbf { f } + \mathbf { g } )$ is continuous at the point $X = 1$, which of the following could be the graph of the function g?
A) [graph A]
B) [graph B]
C) [graph C]
D) [graph D]
E) [graph E]