Pre-Calculus Homework Assignment: Chapter 2 and Radical Functions

This Pre-Calculus assignment solution covers a range of topics, including functions, graphing, and equations. The document provides solutions to multiple-choice questions, graphing problems involving transformations and determining domain and range. It also includes solutions for radical functions, polynomial equations, factoring, and solving for variables. The assignment further explores concepts such as the area of a tennis ball, inverse functions, and sketching graphs. Furthermore, it provides worked solutions for various problems, including factoring cubic and quartic equations and determining the dimensions of a swimming pool. The document is designed to aid students in understanding and solving complex Pre-Calculus problems.

Pre-Calculus
Name:
Institution:
25th July 2018

Answer completion
1. K represents the y-intercept
2. Y values that are negative
3. All the non-negative real values of x
4. X=-11
Chapter 2 Multiple Choice Qs
1. D
2. A
3. C
4. A
5. D
6. C
7. B
8. C
9. C
10. D
11. C
12. B
13. B
14. B
15. C
16. D
17. D

18. D
19. B
20. A
Part 2-Graphing
1. Sketch both f (x) and g(x ) on the graphs below if g ( x )=f ( x−1 ) +2
a. f ( x )= √x
Solution
g ( x ) = √ x−1+2
b. f ( x )=− √x
Solution
g ( x )=− √ x−1+2

2. Determine the domain and range of each function
Equation Domain Range
a. f ( x )=2 √−x +3 { x ∈ R : x ≤ 0 } { y ∈ R : y ≥ 3 }
b.
g ( x )=2 √ 2
3 ( x−2 )+2
{ x ∈ R : x ≥ 2 } { y ∈ R : y ≥ 2 }
c. y−2=− √ 3(x +1) { x ∈ R : x ≥− 5
3 } { y ∈ R : y ≤ 0 }
3. Sketch the graph of each function f ( x ) and then sketch the graph g ( x )= √f ( x ) on the axes
below;
i) f ( x ) =x +2

Equation f ( x) g( x )
Domain Set of all real
numbers
{ x ∈ R : x ≥−2 }
Range Set of all real
numbers
{ y ∈ R : y ≥ 0 }
ii) f ( x ) =x2−4
Solution

Equation f (x) g( x )
Domain Set of all real numbers { x ∈ R : x ≤−2∨x ≥ 2 }
Range { y ∈ R : y ≥−4 } { y ∈ R : y ≥ 0 }
iii) f ( x ) = ( x −2 ) 2+ 4
Solution
Equation f (x) g( x )
Domain Set of all real
numbers
Set of all non-negative real numbers
Range { y ∈ R : y ≥ 4 } { y ∈ R : y ≥ 2 }
4. Solve the following equation algebraically
3 √2 x +4 +9=12

Solution
3 √2 x +4=12−9
3 √2 x +4=3
( 3 √ 2 x + 4 ) 2=32
9 ( 2 x+ 4 )=9
18 x+ 36=9
18 x=−27
x=−27
18 =−3
2 =−1.5
5. Determine the equation of the following radical function in the form y=a √ b( x−h)+k
Solution
From the graph we have;
h=3
k =3
a=3.5
b=1
y= 7
2 √ ( x−3)+3
6. Area of the tennis ball
Solution
r =1
2 √ A
π
r2= 1
4 ( A
π )
A=4 π r2= 4∗22
7 ∗3.32=136.9029 cm2

Section One: Multiple Choice
1. D
2. C
3. A
4. C
5. A
6. C
Section 2:
7. Quiz
Solution
The hshows the shift either upwards or downwards while the k shows the y-intercept
of the curve
8. Quiz
Solution
The value of x must be multiplied with 3 and then the function has to be multiplied by
2
x f(x)
0 0
1 3.46410
2
4 6.92820

3
9 10.3923
9. Quiz
Solution
We have the values as
(8,16) and (4,8)
Gradient is;
Gradient= 16−8
8−4 = 8
4 =2
Thus the equation is;
y−8
x−4 =2
y−8=2(x −4)
y=2 x−8+ 8=2 x
y=2 x
10. Graph of the reflection through x-axis

Solution
11. Filling the table
Equation Words Mapping
y=4 f (x) 4 times the f(x) function 4
y=f ( 3 ( x ) )+ 2 Multiply the f(x) function
by 3 and then add 2
3
y=f (−(x +3)) f(x) function added 3 then
the negative
-1
12. image point
a) (-4,18)
b) (6,25)
c) (13,24)
d) (3,21)
13. Inverse
a) E
b) C
c) B
d) A
e) D

14. Inverse
f ( x )=7 x g ( x )=−3 x+ 4 h ( x )= x+ 4
3 k ( x )= x
3 −5
f−1 ( x )= 1
7 x g−1 ( x )= 4−x
3
h−1 ( x )=3 x +4 k−1 ( x )=3 ( x+5)
15. The equation of the semi-circle is;
Solution
g ( x )=4 f ( 3 ( x−4 ) ) +4
Multiple choice
1. A
2. C
3. B
4. A
5. B
6. D
7. D
8. C
9. B
10. B
11. B

12. A
Short Answer
1. Show that x+1 is a factor of P ( x )=x3+ 2 x2+3 x +2. Explain how you know it is a
factor
Solution
P ( x )=x3+ 2 x2+3 x +2= ( x+1 ) (x2 + x +2)
This means that
P ( x ) = ( x+ 1 ) (x2 +x +2)
Thus ( x +1 ) is a factor of P(x )
2. Factor the equation y=x3−2 x2−5 x+ 6 and then sketch a graph of it
Solution
y=x3−2 x2−5 x+ 6= ( x −1 ) ( x+ 2)( x−3)
Sketch