Functions Problems 2
1. For each of the following graphs and relations between domain x and range y, explain
whether or not it is a function. If it is a function, also explain if it is one-to-one.
(a)
(b)
Nothis isnot a function
This is aone to onefunctionas
as it doesn'tpass
(c)
itpassesbothverticalad horizontal
iieitid
linetests
(d)
This isnot
Thisis afunction 外 not a oneto one
afunctionbecausethere
is morethenIoutputfor1input
si
function he
there is
inputbutmorethan I
onlyloutpatf
every
2. Describe the domain and range of the following functions
using interval
notation.
(a) f (x) = 4x2 → 8x + 4
D
x
x
6x → 9
x
2x → 3
!
"
x
(c) h(x) = log 14 +
5
R 0 x
D 1 到U经 x R 3
(b) g(x) =
D 170 0 R logo 0
1
but
inputcorrespond
tothesamenntp
pb
I
3
1 0
Nreasolul us
4
LT
0
H
0 Positive 1 e Negatneilx.tl
X
x
x
3
3. Let g(x) = x + 1. Determine the zero and the intervals where the function is positive,
xH
negative, increasing, and decreasing. Explain why the function doesn’t have a local
0 0
minimum or maximum. Increasing
Decreasing
DNE
maximum because it isalways
Thefunctiondoesnthave a localminimumor
increasing
2
4. Given the function f (x) = →9 + 4x → 5x , evaluate the following:
(a) f (3 → x)
#
$
2x
(b) f → → 4
5
(c) f (x + h)
iiiiiǒii 蕊
9 4xthl 5lxthi
in
1_Ts ht4x
5. The following piece-wise function models the fee for a table tennis session based on the
number of minutes, t, spent in the session.
f (t) =
$50
t < 30
.tl203
$100
30 ↑ t < 70
$110 + 2(t → 70) t ↓ 70
四
(a) Determine the fee for a 20-minute-long table tennis session.
H60
(b) Determine the fee for an hour-long table tennis session.
f 180 110 2180 70
3
6. Find the intervals at which the function f (x) = →(2x+3) →14 is negative and decreasing.
7. Graph the following piece-wise function: 1
0
(c) Determine the fee for a three-hour-long table tennis session.
2
f (x) =
→5x + 5
x → 10
8
x<2
2 ↑ t ↑ 18
师是 t > 18
Page 2
I1
8. Given f (x) = →9x2 + 10x + 7 and g(x) = 5x → 2, evaluate the following:
(a) f (→2) + g(4)
Typo
As(f g)(7)
(b)
91251027 7 5 9 2
国
(c) f (g(x))
(d) (g ↔ f )(x)
iiiiiiiiiii.iiiiiiiiiiiiii.ie
噩噩
9. Consider the functions f (x) = 4x2 → 4 and g(x) = 2x → 3. If f (g(a)) = 480, find all
possible values of a.
fx
11
480 462 4 480
ǎi烈427
10. Graph the following functions.
(a) y = 5|x + 2| → 3
(b) y = 4(x + 6)2 → 7
憔
听ì
11. Determine the intervals where the function g(x) = x3 → x is increasing.
to sohe withoutderivitncsUnnecessarilycomplicated
Impossible
12. The floor of a number (denoted ↗x↘) returns the greatest integer less than or equal to x.
On the interval →2 ↑ x ↑ 2, define f (x) = ↗x↘. Write f (x) as a piecewise function.
13. Identify where the function h(x) =
6x2 + 22x → 8
is negative.
3x → 1
H
Ǜ
0 4
8
1
Negahve
20 8
2
14. A parabola defined by y = →x + 4x → 3 is intersected at two points by a line defined
by y =
x
→ 7. Find the distance between the two points of intersection.
2
望
嵳
烾
州
严燕
2 7 820 xi equations碧
奷收 315.兰Convert
7
T.gg form.ㄒ
the following standard-form平
into slope-intercept
For each pair
of points, find the slope-intercept equation of the line that passes through them. For each
line, write the slope-intercept equations of a line parallel to it and a line perpendicular
to it.
x
12
→ 3y =
6
17
≃
≃
(b) →x + 2 · y = 3 3
(a)
(c) (3, 5) and (→1, 6)
(d) (13, 2) and (4, →5)
1s
7
三
爨
爨
蘁
㸑
爨
Page 3
14
Be morespecific
Dowedothis forland
int n_n
5 0
fla 9a3_3at15a 352 30 9a3_3a 15a 5 0 引 713aN_N
real
16. Suppose f (x) = 9x3 → 3x2 + 15x → 35. If f (a) = →30, find all possible values of a.
o
solutions
4x → 4
17. What value does the function f (x) = 2
approach (get closer to) as x approaches
x →1
1?
2
器
前
告
回
18. Determine whether the following functions are odd, even, or neither.
4
(a) f (x) = x4 → x8
7
(b) g(x) =
(c) h(x) =
Even
x2 → 2x
x3 → x + 4
x
x2 + 1
Neither
Odd
(d) A(t) = 16ωt77 → 4t13 + t211
Odd
19. For each of the following functions, find its inverse. Then, find the domain and range of
both the original and inverse functions:
(a) y = sin x + 2, where →
ω
ω
↑x↑ .
2
2
(b) y = 2 + |x → 4|, where x < 1
)
7
(c) y = 5
→ 12
2x
㸑 品品品
iii
蘁
20. Find the area of the shape bounded by the y-axis, y = 3 +
iiiiiiiiiiiii
25 10
5
进
年 巺
Page 4
≃
25 → x2 , x = 5, and y = 1.
0
