CRESCENT SCHOOL

Advanced Functions Test 2

Name:

PART A

Only NON-GRAPHING calculators are allowed.

I. (a) Complete the following statements: [IK each]

(i) /is an even function if /(-x) = ·ffx)

(ii) f is an odd function if /(-x) = - Tu)
(b) Give an example ofan even function . [IA]

(c) Give a non-polynomial example ofan odd function. [IA]

(d) Give an example ofa function which is neither even nor odd. [IA]

(e) Give an example of a function which is both even and odd. [ II]

I
2. Let /(x)=2021x and g(x)=--. Find (fog)(x). [2A]
2021x

-=- '2. o'2.. I •_.....__

'21> 2. l K

- ...l.
- X
 I
3. Consider the functions f(x) = - and g(x) = x2 - 4 .
x-2

(a) Write down, if there are any, the x-intercept(s) of the graphs of y =fi.x) andy = g(x). [2A]

ffi) := 1--1 C1,.~ 1'10 )("-lt'l,1-e "Gl,p'{\

tJ ()(. I = ·,l-A- hC\ X'- Ol.fITT :1. a.

(b) Write down the domain and range of each of the functions/and g. [4A]

0 O't"l,A,I'\ of f -= ' { "l} , Rt\~ 4 .f : IR'{()}

o~,\-\ o-t !) = rR , f?u~ o:? 1 :=: f 1f ti< I I.fl -4 r

(c) Let p(x) = f(x) x g(x) . Write down the domain and range of p. [3A]
-4

j;){Jfl'lt:t,~~ 1R,f-i.}
Ru~ -= (R., t4}

(d) Let h(x) = (f O g)(x). Find the domain of h. [2A]

1-\()()-= f cq ci:,)
::: f (,<1--- 4)
 x-3 2x-3
4. (a) Solve - - = - - . [3A]
2x+ 1 3x+ 1

(:i<-:>) ('3-,<4 I) -= (-2.<-3) {'2;(-t I)

31'~ BK -3 -::: 4x2:- 4-K- 3

0-:: X (><~4)

x = C Cf( -4

x-3 2x-3
(b) Solve--$;--. [3A]
2x+ 1 3x+1

(2;<.--t () l3,ct I ) [ (3,-<.+ 1) l,<- "3,) - (2J<- ,.) (2.)<"+f )] $ o

(i,c-t() (:3,<-t() (-}(''2- 4,-J ~\)

(2.ie-t, > ('!:,>C-t •) · ?< (1".-+ 4) (:)

)(-Mrerl'OnR . _:l.2 _ _L> ' -4·

- ,, •• r I
 I
5. In each of the following, the graph of y = f(x) is given. Sketch the graph of y = - - on the same
f(x)
system of axes, indicating any vertical asymptotes. (2A each]

(a)

(b)
Name:

PARTB

Only NON-GRAPHING calculators are allowed.

6. Consider the graph of y = f(x) shown in the following figure.

y

... ·.......·.......·.. ·· f 2+·····'·······'·······i······••;-••····•·· y = f(x)

····+·····• •i• ·· .. i··· .. ··/ ····+ 3-·+ ·····•·······•·······•· ······!·······c··
···f-4 + ···•·•·······'·······•·······'······
... J.... ) ..... ) .......

Let g(x) = -2x and h(x) = x- 2 . Please use a ruler when you answer the following.

(a) Sketch the graph of y =(g o f)(x) . [IA]

y

.,,... ....,. . ....,. . ., = f(x)

 (b) Sketch the graph of y = (h o f)(x) . [IA]

y = f(x)

(c) Sketch the graph of y = (f o g)(x) . [IA]

y = f(x)

(d) Sketch the graph of y = (f o h)(x ) . [ IA)

' Y = f(x)
 (e) Sketch the graph of y = J(g of o h)(x)/ . [I I]

y
l1<fch{.()))I -= l--:zf:(,c- 7 )!
-= 2 I Rie- I
2)

I
7. Let f(x) =- .
1-x

(a) Find f(f(x)) and f(f(f(x))) . Simplify your answers. [3A]

frfto) -:.. F( ~ ) f<f(fu1)) f(- ?- +1)

= l -I ..1., =
I- (-~-+1)
1- x:

= _J.:!_ = ..i.
1-Jt - l -,(_

-,-x
-x
-=: X

. l.-t1
,c

2 3
(b) Let / \x)represent f(f(x)) , / >(x)represent /(/(f(x))) , andingeneral / "\x)represent/
composing with itself n times, where n is a positive integer greater than I. Write down an
202 1
expression for / >(x) . [31]

F<'.>~> x -2) r<410<)-:. f(,,() ·-2J ,rc.,>0<>-= fr~~> , ~ tc .

2c2t lkt\v't S Oi :2. vJ,,v,,, ol~d t,'1

f •(29'2,I)Go := f , ('2.)
<.x >

= - -l +l
""
8. Let.l(x) = ax2 +bx+ c. Ifj(x + 2021) = (x + 2000)2 + x + 21 , find a + b + c. (Hint: Replace x by an
appropriately chosen number.) [41)

Cl i: b-t C :=: f [1)

-,:: .f (- '2c·-i.o -t 2c2 1)
't
-:::- (-202(H·'20M)- 202O-t-'.2.f

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