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3 years ago

9

Show that f(x)=√x-5 is one-to-one and find f^-1

1 answer:

ddd [48]3 years ago

8 0

That is your answer and can you please answer mine

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WHAT IS THE REMAINDER WHEN <img src="https://tex.z-dn.net/?f=32%5E%7B37%5E%7B32%7D%20%7D" id="TexFormula1" title="32^{37^{32} }"

Feliz [49]

Recall Euler's theorem: if $\gcd(a,n) = 1$ , then

$a^{\phi(n)} \equiv 1 \pmod n$

where $\phi$ is Euler's totient function.

We have $\gcd(9,32) = 1$ - in fact, $\gcd(9,32^k)=1$ for any $k\in\Bbb N$ since $9=3^2$ and $32=2^5$ share no common divisors - as well as $\phi(9) = 6$ .

Now,

$37^{32} = (1 + 36)^{32} \\\\ ~~~~~~~~ = 1 + 36c_1 + 36^2c_2 + 36^3c_3+\cdots+36^{32}c_{32} \\\\ ~~~~~~~~ = 1 + 6 \left(6c_1 + 6^3c_2 + 6^5c_3 + \cdots + 6^{63}c_{32}\right) \\\\ \implies 32^{37^{32}} = 32^{1 + 6(\cdots)} = 32\cdot\left(32^{(\cdots)}\right)^6$

where the $c_i$ are positive integer coefficients from the binomial expansion. By Euler's theorem,

$\left(32^{(\cdots)\right)^6 \equiv 1 \pmod9$

so that

$32^{37^{32}} \equiv 32\cdot1 \equiv \boxed{5} \pmod9$

7 0

2 years ago

How do you do conditional statements

Goshia [24]

Example P: I do my homework
P: I get my allowance

8 0

3 years ago

What is the correct solution to (x+2)(x+5)

elena-s [515]

Answer:

x^2+7x+10

Step-by-step explanation:

multiply the first by first and first by second

multiply the second by first and second by second

7 0

3 years ago

If in a right triangle, the given legs are a = 10 and b=16, then by using the Pythagorean Theorem find the hypotenuse, c.

Reika [66]

Answer:

$c=2\sqrt{89}$

Step-by-step explanation:

Pythagorean theorem is:

$c^{2} =a^{2} +b^{2}$

$c^{2}=10^{2}+16^{2} \\c^{2} =100+256\\c^{2}=356\\c=\sqrt{356} \\c=2\sqrt{89}$

3 0

3 years ago

How many solutions over the complex number system does this polynomial have?

erastova [34]

5 solutions.

The number of complex solutions (roots) of a polynomial function is the degree of the polynomial.

5 0

3 years ago