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MA 161: Chapter 1 - Functions Review, Study notes of Algebra

A review sheet for Chapter 1 of MA 161, focusing on functions. It includes true or false questions, simplification problems, equation solving, and finding inverse functions. It also covers algebraic concepts such as finding the domain, function composition, and finding equations of lines.

What you will learn

  • What is the inverse function of f(x) = โˆš2x โˆ’ 5?
  • What is the domain of the function f(x) = (x + 2)(x โˆ’ 3)?
  • Solve the equation 7 โˆ’ 2ln(x) = 5.

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2021/2022

Uploaded on 09/27/2022

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MA 161 Review โ€“ Chapter 1 - Functions Name________________
1. True or False?
FALSE ln(๐‘ฅ2+3๐‘ฅ+2)=ln๐‘ฅ2+ln3๐‘ฅ+ln2
TRUE ln(๐‘ฅ2+3๐‘ฅ +2)=ln(๐‘ฅ+2)+ln(๐‘ฅ+1)
TRUE 2๐‘ฅ=๐‘’๐‘ฅln2
TRUE โˆš๐‘ฅ+3 =(๐‘ฅ+3)1/2
FALSE 1
๐‘ฅ4=๐‘ฅโˆ’1/4
FALSE โˆš๐‘ฅ2+1=๐‘ฅ+1
TRUE โˆš๐‘ฅ2+4๐‘ฅ+4 = ๐‘ฅ+2
FALSE 1
๐‘ฅ2+4 =1
๐‘ฅ2+1
4
2. โ€œSimplifyโ€
b. 1
(4
5)โˆ’2 =16
25
f.
(2๐‘ฅ๐‘ฆ2
5๐‘Žโˆ’1๐‘โˆ’1)โˆ’1 =5
2๐‘Ž๐‘๐‘ฅ๐‘ฆ2
3. Solve each equation.
(a) ๐‘’๐‘ฅ=๐‘’๐‘ฅ2โˆ’2 โ†’ ๐‘ฅ=โˆ’1, 2
(b)
4(3๐‘ฅ)=20 โ†’ ๐‘ฅ = ln5
ln3
(d) 7โˆ’2๐‘’๐‘ฅ= 5 โ†’ ๐‘ฅ = 0
(h) ๐ฅ๐จ๐ ๐Ÿ(๐’™โˆ’๐Ÿ“)= ๐Ÿ‘ โ†’ ๐‘ฅ = 13
(i) ๐ฅ๐ง(๐’™โˆ’๐Ÿ“)= ๐Ÿ‘ โ†’ ๐‘ฅ = ๐‘’ 3+5
4. Find ๐‘ฅ.
a. log232 = ๐‘ฅ โ†’ ๐‘ฅ = 5
b.
log21
4=๐‘ฅ โ†’ ๐‘ฅ = โˆ’2
c.
log2โˆš2
3=๐‘ฅ โ†’ ๐‘ฅ = 1
3
d.
log2โˆš4
3=๐‘ฅ โ†’ ๐‘ฅ = 2
3
pf3
pf4
pf5

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MA 161 Review โ€“ Chapter 1 - Functions Name________________

1. True or False?

FALSE ln(๐‘ฅ 2 + 3๐‘ฅ + 2) = ln ๐‘ฅ 2 + ln 3๐‘ฅ + ln 2

TRUE ln(๐‘ฅ 2 + 3๐‘ฅ + 2) = ln(๐‘ฅ + 2) + ln(๐‘ฅ + 1)

TRUE 2 ๐‘ฅ^ = ๐‘’ ๐‘ฅ ln 2

TRUE โˆš๐‘ฅ + 3 = (๐‘ฅ + 3)1/

FALSE

1 ๐‘ฅ 4

FALSE โˆš๐‘ฅ 2 + 1 = ๐‘ฅ + 1

TRUE โˆš๐‘ฅ 2 + 4๐‘ฅ + 4 = ๐‘ฅ + 2

FALSE

1 ๐‘ฅ 2 +

1 ๐‘ฅ 2

1 4

2. โ€œSimplifyโ€

b.

f.

5๐‘Ž โˆ’1^ ๐‘ โˆ’^

โˆ’

3. Solve each equation.

(a)

๐‘’ ๐‘ฅ^ = ๐‘’ ๐‘ฅ^

(^2) โˆ’

(b)

4(3 ๐‘ฅ^ ) = 20 โ†’ ๐‘ฅ =

ln 5

ln 3

(d)

7 โˆ’ 2๐‘’ ๐‘ฅ^ = 5 โ†’ ๐‘ฅ = 0

(h)

(i)

4. Find ๐‘ฅ.

a.

log 2 32 = ๐‘ฅ โ†’ ๐‘ฅ = 5

b.

log 2

c.

log 2 โˆš

3

d.

log 2 โˆš

3

MA 161 Review โ€“ Chapter 1 - Functions Name________________

e.

log 2 ๐‘ฅ = โˆ’3 โ†’ ๐‘ฅ =

f.

log 2 ๐‘ฅ = 4 โ†’ ๐‘ฅ = 16

g.

log 2 ๐‘ฅ =

h.

log 2 ๐‘ฅ = โˆ’

3

5. Find f โˆ’1^ (๐‘ฅ) โ€“ or explain why it doesnโ€™t exist.

(a)

๐‘“ โˆ’1^ (๐‘ฅ) =

(b)

๐‘“ โˆ’1^ (๐‘ฅ) = ๐‘ฆ =

  1. For the given function, find (and simplify) (i) f (x + h) (ii) f (x + h) f (x) (iii) f^ (x+h h)f^ (x)

(a) f (x) = 3x + 7

f (x + h) = 3(x + h) + 7 = 3x + 3h + 7

f (x + h) f (x) = (3x + 3h + 7) (3x + 7) = 3h f (x + h) f (x) h

3 h h

(b) f (x) = x 2

f (x + h) = (x + h) 2 = x 2 + 2xh + h 2 f (x + h) f (x) = (x 2 + 2xh + h 2 ) (x 2 ) = 2xh + h 2 f (x + h) f (x) h

2 xh + h 2 h

h(2x + h) h = 2x + h

(c) f (x) = (^) x^1 +

f (x + h) =

(x + h) + 3

x + h + 3

f (x + h) f (x) =

x + h + 3

x + 3

(x + 3) (x + h + 3)(x + 3)

(x + h + 3) (x + h + 3)(x + 3)

=

(x + 3) (x + h + 3) (x + h + 3)(x + 3)

h (x + h + 3)(x + 3)

f (x + h) f (x) h

h (x+h+3)(x+3) h

h (x+h+3)(x+3) h 1

h (x + h + 3)(x + 3)

h

=

(x + h + 3)(x + 3)

a few basic algebra reminders....

  1. Find an equation of the line that satisfies the given conditions: (a) passes through (-1,-1) and (3, 7) Answer: y = 2x + 1.

(b) passes through (7, 2) and (5, 2) Answer: y = 2.

(c) passes through (-1, 3) and (-1, 5) Answer: x = 1.

(d) passes through (-2, -6) and is parallel to y = 2x + 3. Answer: y = 2x 2.

(e) passes through (4, 2) and is perpendicular to y = 2x + 3. Answer: y = 12 x + 4.

  1. Simplify the expression below (no negative exponents, no compound fractions). (a) โœ“ 3 y

y 2 4

y 3

y 2

y 2

y 4

y 7 (b) 5 x ^2 ( 2 y 0 ) 3 = 5

x 2

x 2

x 2 (c) 2 x+ 3 x 2

x + 2

x 2 3

2(x 2) 3(x + 2)

2 x 4 3 x + 6

(d) x+ p^3 x 2 + 16

x+ p^3 x 2 + 1

x + 4 3

p x 2 + 16

x + 4 3

p x 2 + 16

  1. Combine into a single logarithmic term. (a) ln(x + 2) ln(x 1) = ln

x + 2 x 1

(b)

ln(x + 2) ln(x 1) + ln(x + 1) = ln

x + 2 x 1

  • ln(x + 1) = ln

(x + 2)(x + 1) x 1