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ruslelena [56]
2 years ago
6

Find f(g(x)). State the domain of the composite function. f(x) = 3x-2/x+1 g(x) = x+5/2x-3

Mathematics
1 answer:
AnnyKZ [126]2 years ago
7 0

Answer:

ℝ - {(-2/3),(3/2)}

Step-by-step explanation:

We want the domain of f(g(x)). So, firstly, we have to find the domain for g(x) and, then, for f(g(x)).

- Domain of g(x): Since the expression is a fracion, we must exclude the values of x that make null the denominator. Hence,

g(x)=\dfrac{x+5}{2x-3}\Longrightarrow 2x-3\neq 0\iff \boxed{x\neq\dfrac{3}{2}}

- Domain of f(g(x)): We'll find its expression:

f(x) = \dfrac{3x-2}{x+1}\\\\f(g(x)) = \dfrac{3g(x)-2}{g(x)+1}\\\\f(g(x)) = \dfrac{3\cdot\dfrac{x+5}{2x-3}-2}{\dfrac{x+5}{2x-3}+1}=\dfrac{~~~\dfrac{3(x+5)-2(2x-3)}{2x-3}~~~}{\dfrac{(x+5)+(2x-3)}{2x-3}}\\\\f(g(x)) =\dfrac{3(x+5)-2(2x-3)}{(x+5)+(2x-3)}=\dfrac{3x+15-4x+6}{x+5+2x-3}\\\\\boxed{f(g(x)) =\dfrac{21-x}{3x+2}}

Now, once again, we have to exclude the values of x that make the denominator equals to zero. Thus,

f(g(x)) =\dfrac{21-x}{3x+2}\Longrightarrow 3x+2\neq0\iff \boxed{x\neq-\dfrac{2}{3}}

Lastly, we may write the domanin of f(g(x)):

D(f(g(x)) = \left]-\infty,-\dfrac{2}{3}\right[\cup\left]-\dfrac{2}{3},\dfrac{3}{2}\right[\cup\left]\dfrac{3}{2},\infty\right[

or, just writing in a shorter way:

\boxed{D(f(g(x)) = \mathbb{R}-\left\{-\dfrac{2}{3},\dfrac{3}{2}\right\}}

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gulaghasi [49]

Answer:

153 ? im not sure srry if wrong

Step-by-step explanation:

20 + 7 = 27

180- 27 = 153

6 0
2 years ago
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Ashley found 78 rocks on Monday. on Tuesdays Devin found 45 rocks how many did the find both days
scoundrel [369]
Lets look at the info we have:-

Ashley found 78 rocks on Monday.
Devin found 45 rocks on Tuesday. 

Question:-

How many did they find both days?

The key word for this is "both". So we need to add here. Lets add:-

78 + 45 = 123

So, Ashley and Devin together found 123 rocks.

Hope I helped ya!! xD
3 0
3 years ago
Write the equation of the line graphed below in slope-intercept
Gelneren [198K]
B= y-intercept
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3 0
3 years ago
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Enter the coefficients of the fifth Taylor polynomial T5(x) for the function f(x) = x5−3x4+2x2+5x−2 based at b=1. T5(x)= + (x−1)
DENIUS [597]

Compute the necessary values/derivatives of f(x) at x=1:

f(1)=3

f'(1)=2

f''(1)=-12

f'''(1)=-12

f^{(4)}(1)=48

f^{(5)}(1)=120

Taylor's theorem then says we can "approximate" (in quotes because the Taylor polynomial for a polynomial is another, exact polynomial) f(x) at x=1 by

T_5(x)=\dfrac3{0!}+\dfrac2{1!}(x-1)-\dfrac{12}{2!}(x-1)^2-\dfrac{12}{3!}(x-1)^3+\dfrac{48}{4!}(x-1)^4+\dfrac{120}{5!}(x-1)^5

T_5(x)=3+2(x-1)-6(x-1)^2-2(x-1)^3+2(x-1)^4+(x-1)^5

###

Another way of doing this would be to solve for the coefficients a,b,c,d,e,g in

f(x)=a+b(x-1)+c(x-1)^2+d(x-1)^3+e(x-1)^4+g(x-1)^5

by expanding the right hand side and matching up terms with the same power of x.

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2 years ago
A video store charges $8 to rent a video game. You must be a member to rent from the store, but
mote1985 [20]

Answer:

10 is the amount of video game you need to rent to equal each other

Step-by-step explanation:

Video store A is 8x

Video store B is 3x+50

8x = 3x +50

5x = 50

x= 10

6 0
2 years ago
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