Mathematics, 02.07.2019 15:30, miahbaby2003p2af1b

Need with the composition of two rational functions. screenshot is below

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answered: johnlecona210

We just have to write one function inside the other:

$(f\circ g)(x) = f(g(x))\\\\(f\circ g)(x) = f\left(\dfrac{13}{x+3}\right)\\\\(f\circ g)(x) = \dfrac{2}{\left(\dfrac{13}{x+3}\right)+3}\\\\(f\circ g)(x) = \dfrac{2}{~~\dfrac{13+3(x+3)}{x+3}~~}\\\\(f\circ g)(x) = \dfrac{2(x+3)}{13+3(x+3)}\\\\\boxed{(f\circ g)(x) = \dfrac{2x+6}{3x+22}}$

Since we have a fraction with x in the denominator, we have to remove from the domain the value of that makes null the denominator. Thus,

$3x+22\neq0\Longrightarrow 3x\neq22\Longrightarrow x\neq \dfrac{22}{3}$

Furthermore, we also have to remove the value of x that annuls the denominator of g(x), because that's the first function that we use. So:

$x+3\neq0\Longrightarrow x\neq-3$

Hence, the domain is:

$D((f\circ g)(x)) = \left]-\infty,-3\right[\cup \left]-3,\dfrac{22}{3}\right[\cup\left]\dfrac{22}{3},\infty\right[$

answered: Guest

part a

answer: a reasonable domain is $0 \le n \le 10$ where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

answered: Guest

answer: the correct answer is -32

step-by-step explanation: