Question

Donnie is saving up money for a down payment on a motorcycle. He currently has $4086, but knows he can get a loan at a lower interest rate if he can put down $4962. If he invests the $4086 in an account that earns 5.9% annually, compounded monthly, how long will it take Donnie to accumulate the $4962? Round your answer to two decimal places, if necessary.

Answer 1

@vocaloid

Answer 2

@darkknight

Answer 3

Compound interest formula

A = P(1 + r/n)^(nt)

P is your principal (4086), r is interest rate (0.059), n is the number of times compounded per year (it compounds monthly, so n = 12), A is the final amount (4962) and t is time. plug in and solve for t.

Answer 4

A = P(1 + r/n)^(nt)

4962 = 4086(1 + 0.059/12)^(12t)

right ?

Answer 5

yes

Answer 6

t = ln(827/681) / 12ln (1.004916)

Answer 7

4962/4086 = (1 + 0.059/12)^(12t)

take the ln of both sides

ln(4962/4086) = ln((1 + 0.059/12)^(12t))

bring the 12t down

ln(4962/4086) = 12t * ln((1 + 0.059/12)

divide both sides by 12 * ln((1 + 0.059/12)

t = ln(4962/4086) / (12 * ln((1 + 0.059/12)) = about 3.30

Answer 8

Consider the following functions.

\[f(x) = \sqrt[3]{x}\] and \[g(x) = \sqrt{x-1}\]

Find the formula for (f+g)(x) and simplify your answer. Then find the domain for (f+g)(x). Round your answer to two decimal places, if necessary.

Answer 9

add the two functions (it just ends up being cube root x + sqrt(x-1), you can't really simplify further than that)

for the domain, there's no restrictions on the cube root part, but the part inside the square root must be at least 0

Answer 10

f(sqrtx-1) = 6sqrtx-1\[f({\sqrt{x-1}}) = \sqrt[6]{x-1}\]

Answer 11

you cannot combine different roots like that

it's just f(x) + g(x) = \[\sqrt{x}+\sqrt[3]{x-1}\]
that's it, you cannot combine a square root and a cube root

Answer 12

Domain : [1, ∞)

Answer 13

oh i see

Answer 14

domain is correct

Answer 15

*correction on my previous work

Answer 16

I swapped the roots

it's just f(x) + g(x) = \[\sqrt{x-1}+\sqrt[3]{x}\]

Answer 17

Find the formula for (f/g)(x) and simplify your answer. Then find the domain for (f/g)(x). Round your answer to two decimal places, if necessary.

Answer 18

it's the same problem

Answer 19

\[f(x) = \sqrt[3]{x}\] and \[g(x) = \sqrt{x-1}\]

so (f/g)(x) is simply \[\frac{ \sqrt[3]{x} }{ \sqrt{x-1} }\], that's really as far as you can simplify due to the different roots

for domain: remember two things: the square root cannot be negative, **and** the denominator cannot be 0. be sure to consider both factors when determining your domain.

Answer 20

Domain : [1, ∞)

Answer 21

not quite

because the denominator cannot be 0, that means x cannot be 1

therefore, (1, infinity)
with an open bracket around the 1

Answer 22

oh okay

Answer 23

Find (f∘g)(−2) for the following functions.

f(−13)=11 and g(−2)=−13

Answer 24

(f∘g)(−2) means f(g(-2)), which means we take the value of g at -2, then we take that value and plug it into f(x)

g(-2) = -13
so f(g(-2)) = f(-13), which the problem already says f(-13) = 11, so (f∘g)(−2) = 11

Answer 25

Consider the following functions.

f={(−4,0),(1,−1),(−3,4)}
and

g={(−2,1),(0,2),(1,−4)}

Find (f+g)(1).

Answer 26

f(1) means the value of f for x = 1
let's look at the list of points in f={(−4,0),(1,−1),(−3,4)}

(1,-1) is the point we want, since this is where x = 1. (1,-1) means x = 1 and f(x) = -1.
so f(1) = -1.

now, using this same logic, find g(1) by looking at the list of points in g.

then add f(1) and g(1).

Answer 27

g(1) = 2

Answer 28

hmm not quite, let's look at the points in g

g={(−2,1),(0,2),(1,−4)}

since we want g(1), we look at (1,-4), which means g(1) = -4 not 2

so if f(1) = -1
and g(1) = -4
then (f+g)(1) = -1 - 4 = -5

Answer 29

Consider the following functions.

f={(1,−1),(−3,−2)}
and

g={(−2,0),(0,1),(1,3),(4,−2)}

Find (f−g)(1).

Answer 30

f(1) = -2
g(1) = -2

Answer 31

f contains the point (1,-1) so f(1) = -1
g contains the point (1,3) so g(1) = 3

so f(1) - g(1) = -1 - 3 = -4

Answer 32

Consider the following functions.

f={(1,−1),(−3,−2)}
and

g={(−2,0),(0,1),(1,3),(4,−2)}

Find (fg)(1).

Answer 33

f(1) = -1
g(1) = 3

Answer 34

if i'm not mistaken

Answer 35

yup
so for (fg)(1) you would just multiply them, (-1)(3) = -3

Answer 36

Consider the following functions.

f={(1,−1),(−3,−2)}
and

g={(−2,0),(0,1),(1,3),(4,−2)}

Find (f/g)(1).

Answer 37

i believe the answer is -2 ?

Answer 38

(f/g)(1) means divide f(1)/g(1)

f(1) = -1
g(1) = 3

so (f/g)(1) = -1/3

Answer 39

Given f(x), find g(x) and h(x) such that f(x)=g(h(x)) and neither g(x) nor h(x) is solely x.

\[f(x) = \frac{ 2 }{ -x+3 }\]

g(x) =

h(x) =

Answer 40

when you have a function f(x) = a/b (one part divided by another part)

you can make h(x) the bottom part (-x+3)
and g(x) can be the top part/x (so 2/x)

that way, when you take g(h(x)), you'll get the top part/bottom part

Answer 41

3, 0

Answer 42

you're looking for functions here. g(x) and h(x) are functions, not just numbers.

like I said, when you have a function f(x) = a/b (notice how we have f(x) = 2/(-x+3), so it fits the pattern

we make h(x) = -x + 3
and g(x) = 2/x

so that when we take g(h(x)), we get 2/(-x+3)

Answer 43

oh ok

Answer 44

Find (f∘g)(4) for the following functions.

\[f(x) = 4x - 4 \] and \[g(x) = x^2 + 1\]

Answer 45

(f∘g)(4) is a composite function. calculate g(4) by plugging in x = 4 into g(x).

then take the result from that, plug it into f(x)

Answer 46

16

Answer 47

check your arithmetic again

g(x) = x^2 + 1
so g(4) = 4^2 + 1 = 17

from there, f(17) = 4(17) - 4 = 64

Answer 48

Consider the following functions.

\[f(x) = \sqrt{x+5}\] and \[g(x) = \frac{ x-4 }{ 3 }\]

Find the formula for (f∘g)(x) and simplify your answer. Then find the domain for (f∘g)(x). Round your answer to two decimal places, if necessary.

Answer 49

Domain : [-11, ∞)

Answer 50

ok, that's right, what about the formula for (f∘g)(x)

Answer 51

\[\sqrt{x+5} + \sqrt{x-4}\]

Answer 52

i think i'm wrong

Answer 53

take g(x) and plug it into f(x)


\[f(x) = \sqrt{x+5}\] and \[g(x) = \frac{ x-4 }{ 3 }\]
plugging (x-4)/3 into f(x), we get

\[\sqrt{\frac{ x-4 }{ 3 }+5}\]
\[\sqrt{\frac{ x-4 }{ 3 }+\frac{5*3}{3}}\]
\[\sqrt{\frac{ x-4+15 }{ 3 }}\]
\[\sqrt{\frac{ x-11}{ 3 }}\]

Answer 54

it turned out wrong , i try another one

Answer 55

Consider the following functions.

f(x)=1/x and g(x)=x−4/2

Find the formula for (f∘g)(x) and simplify your answer. Then find the domain for (f∘g)(x). Round your answer to two decimal places, if necessary.

Answer 56

Consider the following functions.

f(x)=1/x and g(x)=x−4/2

Find the formula for (f∘g)(x) and simplify your answer. Then find the domain for (f∘g)(x). Round your answer to two decimal places, if necessary.

Answer 57

same logic, take g(x) and plug it into f(x)

Answer 58

\[f(\frac{ x-4 }{2 })\]

Answer 59

you're almost there, now take f(x) = 1/x, replace "x" in the denominator with (x-4)/2

Answer 60

Domain : \[(-\infty , 4) \cup (4,\infty )\]

Answer 61

i almost had it but it failed

Answer 62

\[f(x) = \]

Answer 63

\[f(x)=\frac{ 1 }{ x }\]
we just replace the "x" with (x-4)/2
\[f(x)=\frac{ 1 }{ \frac{ x-4 }{ 2 } }=\frac{ 2 }{ x-4 }\]

Answer 64

ok what do we do here

Answer 65

that's it, you plug in (x-4)/2 into f(x) and that's what you get after simplifying, 2/(x-4)

Answer 66

oh ok, what about the domain up top ?

Answer 67

domain is correct, since we have x - 4 in the denominator, x is all values except 4

Answer 68

your answer was right but the domain wasn't

it was \[(-\infty, 4) \cup (4, \infty)\]

Answer 69

Consider the following functions.

f(x)=1/x and g(x)=x−4/2

Find the formula for (g∘f)(x) and simplify your answer. Then find the domain for (g∘f)(x). Round your answer to two decimal places, if necessary.

Answer 70

yes, (−∞,4)∪(4,∞) is interval notation for "all real values except 4"

Answer 71

anyway for this next problem, (g∘f)(x), take f(x) and plug it into g(x)

Answer 72

oh wait , i think i put it in wrong by mistake by applying the infinity signs twice

Answer 73

1-4x/2x

Answer 74

Domain : (−∞,0)∪(0,∞)

Answer 75

g(x) = (x-4)/2
so we replace the "x" with 1/x

\[g(f(x))=\frac{ \frac{ 1 }{ x }-4 }{ 2 }=\frac{ 1 }{ 2x }-2\]
your domain is correct

Answer 76

it's 1/2x - 2 ?

Answer 77

yes.

Answer 78

Consider the following functions.

f(x) = x^2/3 and g(x) = -4x

Find the formula for (fg)(x) and simplify your answer.

Answer 79

(fg)(x) means multiply the two functions

Answer 80

f(-4x) = (-4)^2/3x^2/3

Answer 81

there's no open circle on this one, you just multiply the two functions, you don't plug one into the other

Answer 82

-4x^5/3

Answer 83

good

x^(5/3) = the cube root of x^5, since these are odd roots there's no restrictions on domain, so the domain is (-infinity, infinity)

Answer 84

they don't ask for the domain

Answer 85

do i enter x^(5/3)

Answer 86

the product is -4x^5/3 as you determined earlier

Answer 87

oh ok

Answer 88

it wasn't right

Answer 89

Consider the following functions.

f(x)=x^2/3 and g(x)=√x−3

Find the formula for (fg)(x) and simplify your answer.

Answer 90

so it is just fg(x), right, no open circle?

Answer 91

\[f(x) = x^{\frac{ 2 }{ 3 }}\]

\[g(x) = \sqrt{x-3}\]

Answer 92

(f/g)(x) =

Answer 93

make sure to include the appropriate sign (addition, multiplication, division, etc.).

(f/g)(x) is f divided by g, so f/g(x) = x^(2/3) / sqrt(x-3). can't really simplify beyond that.

Answer 94

make sure to put sqrt(x-3) in the denominator not just sqrt(-3)

Answer 95

i can't

Answer 96

oh wait

Answer 97

no i can't

Answer 98

... strange. I don't think there's another way to write the function.

Answer 99

the answer was

\[\frac{ x^{{\frac{ 2 }{ 3 }}} }{\sqrt{x-3}}\]

Answer 100

Given f(x), find g(x) and h(x) such that f(x)=g(h(x)) and neither g(x) nor h(x) is solely x.

f(x)=√−x^3+2 - 2

Answer 101

g(x) =

h(x) =

Answer 102

can you take an image of this?

Answer 103

when you have f(x) as two parts (a-b) you can make h(x) the first part (a) and g(x) the second part (x - b)

h(x) = sqrt(-x^3+2)
and g(x) = x - 2

that way, when you put them together, g(h(x)) becomes the original function

Answer 104

Find (f∘g)(1) for the following functions. Round your answer to two decimal places, if necessary.

\[f(x) = \sqrt{x+3}\] and \[g(x) = 4 + \sqrt{x}\]

Answer 105

(f∘g)(1)

calculate g(1) by plugging in x = 1
then take that end result, plug it into f(x)

Answer 106

18

Answer 107

check your arithmetic again

g(1) = 4 + sqrt(1) = 5
f(5) = sqrt(5+3) = sqrt(8) = 2.83

Answer 108

oh that's right , i did it wrong