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3 years ago

F(x)=3x+1 and g(x)=x^2-6, find (f+g)(x)

1 answer:

7 0

Answer: x^2+3x-5

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Work Shown:

(f+g)(x) = f(x) + g(x)
(f+g)(x) = [ f(x) ] + [ g(x) ]
(f+g)(x) = [ 3x+1 ] + [ x^2-6 ]
(f+g)(x) = 3x+1 + x^2-6
(f+g)(x) = x^2 + 3x + (1 - 6)
(f+g)(x) = x^2 + 3x - 5

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riadik2000 [5.3K]

Let

$S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n$

where we assume |r| < 1. Multiplying on both sides by r gives

$r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}$

and subtracting this from $S_n$ gives

$(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}$

As n → ∞, the exponential term will converge to 0, and the partial sums $S_n$ will converge to

$\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}$

Now, we're given

$a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a$

$a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}$

We must have |r| < 1 since both sums converge, so

$\dfrac{15}a = \dfrac1{1-r}$

$\dfrac{150}{a^2} = \dfrac1{1-r^2}$

Solving for r by substitution, we have

$\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)$

$\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}$

Recalling the difference of squares identity, we have

$\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}$

We've already confirmed r ≠ 1, so we can simplify this to

$\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15$

It follows that

$\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12$

and so the sum we want is

$ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}$

which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?

7 0

2 years ago

What is the radius of the unit circle?

Valentin [98]

Here is the answer Buddy
213.23

8 0

3 years ago

Read 2 more answers

What are the steps to factoring 6x^2+5x-6

Tanya [424]

Answer:

there is only one easy way to sole this

using quadratic equation

$\frac{ - b + \sqrt{ {b}^{2} - 4ac} }{2a} \\ and \: \: \\ \frac{ - b - \sqrt{ {b}^{2} - 4ac} }{2a}$

Step-by-step explanation:

a=6 b=5 c=-6

and by applying it to the equation you will get

x=0.66 and x=-1.5

7 0

4 years ago

Mr. Nowling's garden has a length that is 7 feet more then twice the width. Draw a diagram to represent this situation. Label ea

azamat

Answer:

Step-by-step explanation:

the expression for the perimeter would be

2(x) + 2(2x+7)

which can be simplified to 2x+4x+14

which equals 6x+14

8 0

4 years ago

What is 17 less than a number? HELP!

NARA [144]

Answer:

x - 17

Step-by-step explanation:

hope that helps

5 0

3 years ago