The document contains 20 multiple choice questions about functions. The questions cover topics such as the domain and range of functions, function composition, finding maximums and minimums, and determining if functions are injective or surjective. The solutions key indicates that questions 1, 3, 4, 6, 9, 10, 15, 16, and 18 were answered correctly.
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MATH 115 Precalculus Fall, 2018, V1.4
Page 1 of 7
MATH 115 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 28 problems.
Problems #1–6 are Multiple Choice.
Problems #7–17 are Short Answer. (Work not required to be shown)
Problems #18–28 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Solve | 7 – 5x | 8 and write interval notation for the solution set. 1. _______
A. (−∞, −
1
5
] ∪ [3, ∞)
B. (−∞, 3] ∪ [−
1
5
, ∞)
C. [−
1
5
, ∞)
D. [−
1
5
, 3]
2. Which of the following polynomials has a graph which exhibits the end behavior of
upward to the left and upward to the right? 2. ______
A. f (x) = x3 – 3x2 – x
B. f (x) = 3x6 + x2 – x
C. f (x) = –7x4 – x3 – 5
D. f (x) = –6x5 + x3 + 8
3. Write as an equivalent expression: 9 log x – log (y + 3) + log 1 3. _______
A.
9
log
log ( 3)
x
y +
B. ( )log 9 2x y− −
C.
9 1
log
3
x
y
+
+
D.
9
log
3
x
y
+
vvvv
MATH 115 Precalculus Fall, 2018, V1.4
Page 2 of 7
4. Determine the interval(s) on which the function is decreasing. 4. ______
A. (–, –3.5) (0, 3.5)
B. (– 5.3, 5.3)
C. (– 3.5, 3.5)
D. (– 2, 2)
5. Which of the functions corresponds to the graph? 5. ______
A. ( ) 1xf x e−= +
B. ( ) 2xf x e−= +
C. ( ) 2xf x e−=
D. ( ) 2 xf x e−=
MATH 115 Precalculus Fall, 2018, V1.4
Page 3 of 7
6. Which of the functions corresponds to the graph? 6. ______
A. f (x) = 3 – sin x
B. f (x) = sin x + 3
C. f (x) = 2 cos x + 1
D. f (x) = cos(2x) + 2
SHORT ANSWER:
7. Points (–9, 2) and (–5, 8) are endpoints of the diameter of a circle.
(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________
(b) What is the center of the circle? Answer: ____________
(c) What is the equation of the circle? Answer: ___________________________
8. Find the value of the logarithm: log8 (
1
64
). Answer: ____________
9. A salesperso.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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2. FUNÇÕES
1
01. (Espcex 2015) Assinale a alternativa que representa o conjunto de todos os números reais para os quais está
definida a função
2
3 2
x 6x 5
f(x) .
x 4
− +
=
−
a) ℝ − {−2,2}
b) ( ) ( )
, 2 5,
− − +
c) ( ) ( )
, 2 2,1 5,
− − − +
d) ( ) ( )
,1 5,
− +
e) ( )
, 2 2,
− − +
02. (Epcar 2015) Considere a função real 𝑓: ℝ → ℝ definida por x
f(x) a b,
= − em que 0 a 1
e b 1
. Analise as
alternativas abaixo e marque a falsa.
a) Na função ,
f se x 0,
então b f(x) 1 b
− −
b) Im(f) contém elementos menores que o número real b
−
c) A raiz da função f é um número negativo.
d) A função real h, definida por ( )
h(x) f | x |
= não possui raízes.
03. (Espcex 2014) Na figura abaixo está representado o gráfico da função polinomial f, definida no intervalo real [a,b].
Com base nas informações fornecidas pela figura, podemos afirmar que
a) f é crescente no intervalo [a,0].
b) f(x) f(e)
para todo x no intervalo [d, b].
c) f(x) 0
para todo x no intervalo [c, 0].
d) a função f é decrescente no intervalo [c,e].
e) se 1
x [a,c]
e 2
x [d,e],
então 1 2
f(x ) f(x ).
04. (Espcex 2014) Uma indústria produz mensalmente x lotes de um produto. O valor mensal resultante da venda
deste produto é 2
V(x) 3x 12x
= − e o custo mensal da produção é dado por 2
C(x) 5x 40x 40.
= − − Sabendo que o
lucro é obtido pela diferença entre o valor resultante das vendas e o custo da produção, então o número de lotes
mensais que essa indústria deve vender para obter lucro máximo é igual a
a) 4 lotes
b) 5 lotes
c) 6 lotes
d) 7 lotes
e) 8 lotes
3. FUNÇÕES
2
05. (Ita 2014) Considere as funções 𝑓, 𝑔: ℤ → ℝ, 𝑓(𝑥) = 𝑎𝑥 + 𝑚 , 𝑔(𝑥) = 𝑏𝑥 + 𝑛, em que a, b, m e n são
constantes reais. Se A e B são as imagens de f e de g, respectivamente, então, das afirmações abaixo:
I. Se A B,
= então a b
= e m n;
=
II. Se 𝐴 = ℤ, então a 1
;
=
III. Se 𝑎, 𝑏, 𝑚, 𝑛 ∈ ℤ, com a b
= e m n,
= − então A B,
= é (são) verdadeira(s)
a) apenas I
b) apenas II
c) apenas III
d) apenas I e II
e) nenhuma
06. (Acafe 2014) Uma pequena fábrica de tubos de plástico calcula a sua receita em milhares de reais, através da
função R(x) 3,8x,
= onde x representa o número de tubos vendidos. Sabendo que o custo para a produção do mesmo
número de tubos é 40% da receita mais R$ 570,00. Nessas condições, para evitar prejuízo, o número mínimo de tubos
de plástico que devem ser produzidos e vendidos pertence ao intervalo
a) [240 ; 248]
b) [248 ; 260]
c) [252 ; 258]
d) [255 ; 260]
07. (Epcar 2014)Considere os gráficos das funções reais 𝑓: 𝐴 → ℝ e 𝑔: 𝐵 → ℝ. Sabe-se que A [ a, a];
= − B ] , t];
= −
g( a) f( a);
− − g(0) f(0);
g(a) f(a)
e g(x) n
= para todo x a.
−
Analise as afirmativas abaixo e marque a FALSA.
a) A função f é par.
b) Se x ] d, m [,
então f(x) g(x) 0
c) Im(g) [ n, r [ {s}
=
d) A função ℎ: 𝐸 → ℝ dada por
2
h(x)
f(x) g(x)
−
=
−
está definida se 𝐸 = {𝑥 ∈ ℝ| − 𝑎 ≤ 𝑥 < −𝑑 𝑜𝑢 𝑑 < 𝑥 ≤ 𝑎}
08. (Esc. Naval 2014) Considere as funções reais x
100
f(x)
1 2−
=
+
e
x
2
g(x) 2 ,
= 𝑥 ∈ ℝ. Qual é o valor da função composta
1
(g f )(90)?
−
a) 1 b) 3 c) 9 d)
1
10
e)
1
3
4. FUNÇÕES
3
09. (Esc. Naval 2014) Considere a função real de variável real 2 x
f(x) x e .
= A que intervalo pertence à abscissa do ponto
de máximo local de f em ] , [?
− +
a) [ 3, 1]
− −
b) [ 1
,1[
−
c)
1
0,
2
d) ]1, 2]
e) ]2, 4]
10. (Esc. Naval 2014) Sabendo que logx representa o logaritmo de x na base 10, qual é o domínio da função real de
variável real
3
3
x
arccos log
10
f(x) ?
4x x
=
−
a) ]0, 2[
b)
1
,1
2
c) ]0,1]
d) [1, 2[
e)
1
, 2
2
11. (Ita 2014) Das afirmações:
I. Se 𝑥, 𝑦 ∈ ℝℚ, com y x,
− então 𝑥 + 𝑦 ∈ ℝℚ;
II. Se 𝑥 ∈ ℚ e 𝑦 ∈ ℝℚ, então 𝑥𝑦 ∈ ℝℚ;
III. Se 𝑎, 𝑏, 𝑐 ∈ ℝ, com a b c.
Se
f : a,c a,b
→ é sobrejetora, então f não é injetora, é (são) verdadeira(s)
a) apenas I e II
b) apenas I e III
c) apenas II e III
d) apenas III
e) nenhuma
12. (Esc. Naval 2014) Após acionado o flash de uma câmera, a bateria imediatamente começa a recarregar o capacitor
do flash, que armazena uma carga elétrica dada por
t
2
0
Q(t) Q 1 e ,
−
= −
onde 0
Q é a capacidade limite de carga e t
é medido em segundos. Qual o tempo, em segundos, para recarregar o capacitor de 90% da sua capacidade limite?
a) n10
b) 2
n(10)
c) n10
d) 1
( n10)−
e) 2
n(10)
5. FUNÇÕES
4
13. (Esc. Naval 2014) Considere as funções reais
x
f(x) nx
2
= − e 2
x
g(x) ( nx)
2
= − onde nx expressa o logaritmo
de x na base neperiana e (e 2,7).
Se P e Q são os pontos de interseção dos gráficos de f e g, podemos afirmar
que o coeficiente angular da reta que passa por P e Q é
a)
e 1
2(e 3)
+
−
b) e 1
+
c)
e 1
2(e 1)
−
+
d) 2e 1
+
e)
e 3
2(e 1)
−
−
14. (Esc. Naval 2014) Uma bolinha de aço é lançada a partir da origem e segue urna trajetória retilínea até atingir o
vértice de um anteparo parabólico representado pela função real de variável real 2
3
f(x) x 2 3x.
3
−
= +
Ao incidir
no vértice do anteparo é refletida e a nova trajetória retilínea é simétrica à inicial, em relação ao eixo da parábola.
Qual é o ângulo de incidência (ângulo entre a trajetória e o eixo da parábola)?
a) 30
b) 45
c) 60
d) 75
e) 90
15. (Espcex 2013) A figura a seguir apresenta o gráfico de um polinômio P(x) do 4º grau no intervalo
0,5 .
O número de raízes reais da equação ( )
P x 1 0
+ = no intervalo
0,5 é
a) 0
b) 1
c) 2
d) 3
e) 4
6. FUNÇÕES
5
16. (Esc. Naval 2013) Numa vidraçaria há um pedaço de espelho, sob a forma de um triângulo retângulo de lados
30 cm, 40 cm e 50 cm.
Deseja-se, a partir dele, recortar um espelho retangular, com a maior área possível, conforme figura. Então as
dimensões do espelho são
a) 25 cm e 12 cm
b) 20 cm e 15 cm
c) 10 cm e 30 cm
d) 12,5 cm e 24 cm
e) 10 3 cm e 10 3 cm
17. (Esc. Naval 2013) Considere a função real y f(x),
= definida para 5 x 5,
− representada graficamente abaixo.
Supondo a 0
uma constante real, para que valores de a o gráfico do polinômio ( )
2
p(x) a x 9
= − intercepta o gráfico
de y f(x)
= em exatamente 4 pontos distintos?
a)
10
1 a
9
b)
2
a 1
9
c)
2
0 a
9
d)
10
a 3
9
e) a 3
18. (Esc. Naval 2013) A reta no ℝ2
de equação 2y 3x 0
− = intercepta o gráfico da função
2
x 1
f(x) x
x
−
= nos pontos
P e Q. Qual a distância entre P e Q?
a) 2 15
b) 2 13
c) 2 7
d) 7
e)
5
2
7. FUNÇÕES
6
19. (Ita 2013) Considere funções f, g, 𝑓 + 𝑔: ℝ → ℝ. Das afirmações:
I. Se f e g são injetoras, f g
+ é injetora;
II. Se f e g são sobrejetoras, f g
+ é sobrejetora;
III. Se f e g não são injetoras, f g
+ não é injetora;
IV. Se f e g não são sobrejetoras, f g
+ não é sobrejetora,
é (são) verdadeira(s)
a) nenhuma
b) apenas I e II
c) apenas I e III
d) apenas III e IV
e) todas
20. (Ita 2013) Considere as funções f e g, da variável real x, definidas, respectivamente, por ( )
2
x ax b
f x e + +
= e
( ) ax
g x ln ,
3b
=
em que a e b são números reais. Se ( ) ( )
f 1 1 f 2 ,
− = = − então pode-se afirmar sobre a função composta
g f que
a) ( )
g f 1 ln 3.
=
b) ( )
g f 0 .
c) g f nunca se anula.
d) g f está definida apenas em {𝑥 ∈ ℝ:𝑥 > 0}.
e) g f admite dois zeros reais distintos.
GABARITO
1 - C 2 - B 3 - D 4 - D 5 - E
6 - B 7 - B 8 - B 9 - A 10 - D
11 - E 12 - B 13 - E 14 - A 15 - C
16 - A 17 - C 18 - B 19 - A 20 - E