Pre Calc Chapter 1
illiabastieiev September 2020
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1 reflection
A little more than a third of the problems i was able to do with my prior knowledge but the rest i had to google to look up the rules. for example the transformation problems in a section i didn’t remember the rules to so i had to look up the rules and then i was able to plug it into the equations.
for some of my problems i checked my problems in mathematica and almost all of them were right. The hardest thing for me was organizing my math in overleaf especially when i was using complex fractions. 4.2 and 5.5 there were a lot fractions together and there was a lot of math all over the place which i had to figure out how to organize.
the hardest part of the chapter was 4.5 as i didnt know how to solve the rest of it. the problem above it was way easier as i just plugged in the numbers but for 4.5 there was a variable that i didnt know.
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2 1.1: Functions and Function Notation 2.1 worked example 1 page 17.
For each of the following functions, evaluate: f(−2),f(−1),f(0),f(1),andf(2) 22. f(x)=8−3x
25. f(x) = −x3 + 2x
2.2 Answers:
f(x) = 8 − 3x
f(−2) = 8 − 3(−2) = 8 + 6 = 14 f(−1) = 8 − 3(−1) = 8 + 3 = 11 f(0) = 8 − 3(0) = 8 − 0 = 8 f(1) = 8 − 3(1) = 8 − 3 = 5 f(2) = 8 − 3(2) = 8 − 5 = 3
f(x) = −x3 + 2x
f(−2)=−(−2)3 +2(−2)=8−4=4 f(−1)=−(−1)3 +2(−1)=1−2=−1 f(0)=−(0)3 +2(0)=0+0=0 f(1)=−(1)3 +2(1)=−1+2=1 f(2)=−(2)3 +2(2)=−8+4=−4
2.3 Comments
This problem was pretty easy math wise but this was the first problem i did on overleaf so it was a bit difficult to figure out how to use it.
2.4 worked example 2 page 18
35. Suppose f(x) = x2 + 8x − 4 Compute the following a. f(−1) + f(1).
b.f(−1)−f(1)
2.5 Answers:
f(−1) + f(1) =
((−1)2 +8(−1)−4)+((1)2 +8(1)−4) (1 − 8 − 4) + (1 + 8 − 4) = −11 + 5 = 16
f(−1) − f(1)
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((−1)2 +8(−1)−4)−((1)2 +8(1)−4) (1 − 8 − 4) − (1 + 8 − 4) = −11 − 5 = 6
2.6 comments
this problem was pretty easy as it was just adding 2 functions together.
2.7 worked example 3 page 19
43. write the equation of the circle centered at (3,-9) with radius 6.
2.8 answers:
Formulaofacircle=(x−h)2+(y−k)2 =r2 (x−3)2 +(y+9)2 =62
2.9 comments
i was confused at first because i forgot the formula of a circle but after i googled it and plugged it back in i remembered how simple it is and did it pretty easily
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3 1.2 Domain and Range 3.1 Worked example 1 page 34
Find the Domain of Each Function
√ 7.f(x)=3 x−2
Domain is x is greater than or equal to 2
3.2 Answers
Domain is x is greater than or equal to 2
the Square root can not have a number less than 2 as then it wont be a real number because the square root will have an undefined number
3.3 Comments
this was easy as you just plug in numbers and if its not a negative under the square root then it works
3.4 Worked example 2 page 34
Given each Function, evaluate: f(−1),f(0),f(2),f(4)
19. f(x)=7x+3ifxislessthan0andf(x)=7x+6ifxisgreaterorequal to 0
3.5 Answers
f(−1) = 7x+3 = 7(−1)+3 = −7+3 = −4 f(0) = 7x + 6 = 7(0) + 6 = 0 + 6 = 6
f(2) = 7x + 6 = 7(2) + 6 = 14 + 6 = 20 f(4) = 7x + 6 = 7(4) + 6 = 28 + 6 = 34
3.6 Comments
I liked this one because it was just plugging in the numbers but you have to know which equation to use.
3.7 worked example 3 page 35
do 2 of 31-36 SKETCH GRAPH
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4 1.3 Rates of change behavior of graphs 4.1 Worked example 1 page 48
5,6. Find the average rate of change of each function on the interval specified 5. f(x) = x2 on [1,5]
6. q(x) = x3 on [−4, 2]
4.2 Answers
5. f(x) = x2 on [1,5]
f(1)=(1)2 =1
f(5)=(5)2 =25
f (b)−f (a) f (5)−f (1) 25−1 = 6
b−a 5−1 4
6. q(x) = x3 on [−4, 2]
f(−4) = (−4)2 = 16
f(2)=(2)2 =4
f (b)−f (a) f (16)−f (4) 26−4 =3.667
b−a −4−2 −6 4.3 Comments
this problem was hard because i had to watch a YouTube video to remember how to find the rate of change. also because i had to figure out how to use fractions in overleaf and organize a lot of math.
4.4 worked example 2 page 48
Find the average rate of change of each function on the interval specified. your answers will be expressions involving a parameter (b or h)
11.f (x) = 4×2 − 7 on [1, b]
4.5 answers
11.f (x) = 4×2 − 7 on [1, b]
f(1)=4(1)2 −7=4−7=−3
f(b) = 4(b)2 − 7 f (b)−f (a)
b−a
4.6 Comments
This one i couldn’t figure out, it was very challenging. i simplified it as much as i could and then i didn’t know what to do as instead of a number for the second variable there was a b. i went as far as i could but couldn’t figure it out even after asking a friend
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5 1.4 Composition of Functions 5.1 Worked example 1 page 60
Given each pair of functions, calculate f(g(0)) and g(f(0)) 1.f(x) = 4x + 8, g(x) = 7 − x2
5.2 Answers
f(g(0) = 7−02) = 4(7−02)+8 = 4(7)+8 = 28+8 = 36
g(f(0) = 4(0)+8) = 7−(4(0)+8)2 = 7−(4+8)2 = 7−(12)2 = 7−144 = −137
5.3 Comments
it took me a second to remember how to do composite functions but it was pretty easy in the end as its just plugging in numbers.
5.4 Worked example 2 page 60
For each pair of functions, find f(g(x)) and g(f(x)). Simplify your answers 5.5 Answers
21.f(x)= 1 ,g(x)=7+6 x−6 x
f(g(x))= 1 =1 ( 7 +6)−6 7 xx
f(g(x)) = g(f(x))=
1 f(g(x)) = x 77 x
7 +6=7x−42 1
x−6 g(f(x)) = 7x − 42
22.f(x)= 1 ,g(x)=2+4
x−4 f(g(x))= 1
x
=1 =2 2 x
g(f(x))= 2 +4=2x−4 1
x−4 g(f(x)) = 2x − 4
5.6 Comments
this problem was hard to organize as it was a lot of math to fit in and there were fractions under fractions but in the end i got the answer and was able to organize it.
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f(g(x)) = x 2
2 +4−4 xx
5.7 Worked example 3 page 61
If f(x) = x4 + 6, g(x) = x − 6 and h(x) = √x, find F(g(h(x))) 5.8 Answers
f(g(h(x))) = (((√x) − 6)4) + 6 f(g(h(x))) = ((x2) + 36) + 6
5.9 Comments
this problem was pretty easy as it wasn’t as much math as the previous one and it was kind of just arranging it in the right order and then simplifying
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6 1.5 Transformation of Functions 6.1 Worked example 1 page
√
11.write formula for f(x) = 6.2 Answer
√
11.f(x)+1= x+2 6.3 Comments
x shifted up 1 unit and left 2 units
i had to google the rules of transformations as i forgot how to do it a bit but after finding out it was pretty easy and the rules came back to me
6.4 worked example 2 page 87
33. Starting with the graph of f(x) = 6x write the equation of the graph that results from
a. reflecting f(x) about the x-axis and y-axis
b. reflecting f(x) about the x-axis, shifting left 2 units, and down 3 units
6.5 Answer
a −f(−x) = 6x b.−f(x+2)=62 −3
6.6 Comments
This problem was also easy as it was just reflections and there are certain rules for that.
6.7 Worked example 3 page 88
39. for each equation below, Determine if the function is Odd, Even, or Neither a. f(x) = 3×4
√
b.g(x)= x
c. h(x) = 1 + 3x
x
6.8 Answer
a.f(x) = 3×4
find (f(−x)
f(−x) = 3×4
even if f(−x) = f(x) Equation is even
b.g(x)= x
√
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g(−x) = doesn’t exist
Equation is neither
c. h(x) = 1 + 3x x
f(−x)= −1 −3x x
f(−x) does not equal f(x) Equation is odd
6.9 Comments
I forgot how to figure out weather a function is even, odd, or neither and i had to ask a friend to help me out and do the first problem with me.
6.10 Worked example 4 page 89
67. Determine the Interval(s) on which the function is increasing and decreasing. f(x)=4(x+1)2 −5
6.11 Answer
After putting the equation in the graph i have determined that when x is less than -1, the graph is decreasing and when x is greater than 1, the graph is increasing.
i also know that since (x+1) means that the graphs minimum value where it changes is -1. that means that anything before that goes down and anything after that goes up
6.12 Comments
For this one i had a hard time remembering how to shift it up a unit up. moving it left and right was easier
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7 1.6 Inverse Functions
7.1 Worked example 1 page 100
For each function below, find f−1(x) 13. f(x)=x+3
14. f(x)=x+5
15. f(x)=2−x
16. f(x)=3−x 17. f(x)=11x+7
7.2 Answer
13. f(x)−1. y=x+3x=y+3y=x−3 f(x)−1 = x − 3
14. f(x)−1. y=x+5x=y+5y=x−5 f(x)−1 = x − 5
15. f(x)−1. y=2−xx=2−y−y=x−2y=−x+2 f(x)−1 = −x + 2
16. f(x)−1. y=3−xx=3−y−y=x−3y=−x+3 f(x)−1 = −x + 3
17. f(x)−1. y = 11x + 7 x = 11y + 7 11y = x − 7 y = x−7
f(x)−1 = x−7 11
7.3 Comments
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Finding the inverse was really easy as i just had to switch the x and y and then solve for y.
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