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Yuki888 [10]
2 years ago
7

Let $f(x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equ

ation \[f(x) + f\left(\frac{x-1}x\right) = 1+x.\]find the $f(x)$ satisfying these conditions. write $f(x)$ as a rational function with expanded polynomials in the numerator and denominator.

Mathematics
1 answer:
Mademuasel [1]2 years ago
7 0
We have to find the values of F.
In this case. F is unlikely to be a polynomial.
But the problem is, we can’t calculate the values of F directly.
There is no real value of x for which x = x−1 x because F isn’t defined at 0 or 1. so,
substituting x = 2.
F(2) + F(1/2) = 3.

Substitute, x = 1/2
F(1/2) + F(−1) = −1/2.
We still are not getting the required value,
therefore,
Substitute x = −1

As, F(2) +F(−1) = 0.
now we have three equations in three unknowns, which we can solve.
It turns out that:
F(2) = 3/4
F(3) = 17/12
F(4) = 47/24
and
F(5) = 99/40

Setting
g(x) = 1 − 1/x
and using
2 → 1/2
to denote
g(2) = 1/2
 we see that :
x → 1 - 1/x → 1/(1-x) →x

so that:
g(g(g(x))) = x.

Therefore, whatever x 6= 0, 1 we start with, we will always get three equations in the three “unknowns” F(x), F(g(x)) and F(g(g(x))).
Now solve these equations to get a formula for F(x)

As,
h(x) = (1+x)/(1−x)
which satisfies h(h(h(h(x)))) = x

Now, mapping x to h(x) corresponds to rotating the circle by ninety degrees.

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melamori03 [73]
To calculate expected value,
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3 years ago
Evaluate 8j - k + 14 when j = 0.25 and k = 1.
pantera1 [17]

Steps to solve:

8j - k + 14; j = 0.25 and k = 1

~Substitute

8(0.25) - (1) + 14

~Simplify

2 - 1 + 14

~Subtract

1 + 14

~Add

15

Best of Luck!

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3 years ago
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A projectile is launched from ground level with an initial velocity of v0 feet per second. Neglecting air resistance, its height
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The formula is s=-16t^2+V_ot

We have:
s=64
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Substituting these values into the formula, we have

64=-16t^2+80t

16t^2-80t+64=0 ⇒ divide each term by 16

t^2-5t+4=0 ⇒ find the pair of numbers that multiply to give 4 and sum gives -5 which is -1 and -4

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The time when the projectile reached 64 feet is at t=1 and t=4



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