Vanier College
Department of Mathematics
Sec V Mathematics
201-015-50
Worksheet: Exponential Function
1. Draw the graph of each of the following exponential functions, and analyze each
of them completely.
(1) f (x) = 3 · 2x − 6
x
1
(3) f (x) = −2 ·
+8
2
−x+1
(5) f (x) = −2
+8
−x+1
1
(7) f (x) = −2 ·
−3
2
2. Solve the following equations.
√
(1) 2x · 32 = 8
2x−1
1
(3)
· 9x+3 = 271−x
3
(5)
x2
3
·
1
27
x
x−2
=9
(2) f (x) = −3x + 27
(4)
x
1
f (x) = 5 ·
− 45
3
(6)
−2x−1
1
−6
f (x) = 2 ·
3
−x
1
(8) f (x) = 3 ·
−6
4
(2)
(0.5)x−3 ·
√
3
4 = 8x
x−5
4x+2
2
9
1−3x
(4)
· (1.5)
=
3
4
−x(x+5)
1
(6)
= 42x+3
2
3. Find the equation of the exponential function of the form y = a·cx that contains
the given points.
(1)
(1, 6) and (3, 24)
(2)
(2, −9) and (3, −27)
(3)
(−2, 18) and (−4, 162)
(4)
2
(−1, − ) and (2, −50)
5
(5)
(−1, 2) and (−3, 36)
(6)
(−1, −6) and (−2, −24)
4. 10 000 $ are invested in an account tha yields 6% interest per year. How much
will the account be worth in 20 years if the interest is compounded (1) annually?
(2) semi-annually? (3) monthly? (4) daily?
5. A population of bacteria doubles every hour. If there are 100 bacteria initially,
(1) how many are there after 5 hours?
(2) in how many hours will there be 102 400 bacteria?
6. The population of a city has been growing at a rate of 4% per year. If it has a
population of 100 000 today, what was its population 10 years ago?
7. A sheet, hung out to dry, loses moisture in the wind at a rate of about 60% per
hour. How much moisture will remain after 5 hours?
ANSWERS
(2)
1. (1)
Dom(f ) = R
R(f ) = ] − 6, +∞[
Zeros: 1
Y-intercept: −3
Variation:
f (x) % if x ∈ R
f (x) & if x ∈ ∅
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ [1, +∞[
f (x) ≤ 0 if x ∈] − ∞, 1]
Dom(f ) = R
R(f ) = ] − ∞, 27[
Zeros: 3
Y-intercept: 24
Variation:
f (x) % if x ∈ ∅
f (x) & if x ∈ R
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈] − ∞, 3]
f (x) ≤ 0 if x ∈ [3, +∞[
(3)
(4)
Dom(f ) = R
R(f ) = ] − ∞, 8[
Zeros: −2
Y-intercept: 6
Variation:
f (x) % if x ∈ R
f (x) & if x ∈ ∅
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ [−2, +∞[
f (x) ≤ 0 if x ∈] − ∞, −2]
Dom(f ) = R
R(f ) = ] − 45, +∞[
Zeros: −2
Y-intercept: −40
Variation:
f (x) % if x ∈ ∅
f (x) & if x ∈ R
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈] − ∞, −2]
f (x) ≤ 0 if x ∈ [−2, +∞[
(5)
(6)
Dom(f ) = R
R(f ) = ] − ∞, 8[
Zeros: −2
Y-intercept: 6
Variation:
f (x) % if x ∈ R
f (x) & if x ∈ ∅
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ [−2, +∞[
f (x) ≤ 0 if x ∈] − ∞, −2]
Dom(f ) = R
R(f ) =] − 6, +∞[
Zeros: 0
Y-intercept: 0
Variation:
f (x) % if x ∈ R
f (x) & if x ∈ ∅
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ [0, +∞[
f (x) ≤ 0 if x ∈] − ∞, 0]
(7)
(8)
Dom(f ) = R
R(f ) = ] − ∞, −3]
Zeros: None
Y-intercept: −4
Variation:
f (x) % if x ∈ ∅
f (x) & if x ∈ R
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ ∅
f (x) ≤ 0 if x ∈ R
Dom(f ) = R
R(f ) = ] − 6, +∞[
Zeros: 21
Y-intercept: −3
Variation:
f (x) % if x ∈ R
f (x) & if x ∈ ∅
Extremums: Max: None, Min: None
Sign:
f (x) ≥ 0 if x ∈ [ 12 , +∞[
f (x) ≤ 0 if x ∈] − ∞, 12 ]
7
2
11
(2) x =
12
4
(3) x = −
3
1
(4) x =
6
2. (1) x = −
(5) x = 1 or x = 4
(6) x = −3 or x = 2
3. (1) f (x) = 3 · 2x
(2) f (x) = −3x
x
1
(3) f (x) = 2 ·
3
(4) f (x) = −2 · 5x
x
1
2
(5) f (x) = ·
3
3
x
1
3
(6) f (x) = − ·
2
4
4. (1) 32071.35$
(2) 32620.38$
(3) 33102.04$
(4) 33197.89$
5. (1) 3200 bacteria
(2) 10 hours
6. 67 556
7. 1 % of moisture left.