Algebra Precalculus – Prove the value of a function with odd and even function properties

Question:

given
$$ f (x) + frac {f (-x)} {2} = 2x ,, quad x ne 0 $$

Solve for: $ f (2) $

My work :

To let

$$ a = f (x) $$

$$ b = frac {f (-x)} {2} $$

Consider $ a + b = 2x ,, quad x in Bbb N $

Then there are two options

For case $ a $ is straight and $ b $ is just:

$$ f (-x) = f (x) $$

$$ begin {align}
& 2f (2) + f (2) = 8 \
& f (2) = dfrac {8} {3}
end {align} $$

For case $ a $ is odd and $ b $ is odd:

$$ f (-x) = – f (x) $$

$$ begin {align}
& 2f (2) -f (2) = 8 \
& f (2) = 8
end {align} $$

But my friend told me that she just found a unique solution $ 8 $ using the removal method.

What is true?

Please correct my work. Thank you very much

Algebra pre-calculus – Prove the inequality in the context of Pythagoras theorem

$ sqrt {A ^ 2 + (N * A) ^ 2} + sqrt {B ^ 2 + (N * B) ^ 2} < sqrt {A ^ 2 + (N * A + M) ^ 2} + sqrt {B ^ 2 + (N * BM) ^ 2} $

Where $ A $ and $ B $ are positive real numbers and $ N $ and $ M $ are real numbers.
Enter image description here

$ I + J $ is equal to the left side of the inequality and has the lowest possible value given the length of the lowest segment, the length of $ A $ and the length of $ B $

Algebra Precalculus – How do I check if a function is injective if there are no simple examples?

Suppose I have to check if this function is inverse: $ f: mathbb {R} to mathbb {R} $ defined by $ x (x ^ 2-1) $

A function has an inverse function if it is injective and surjective, i. H. As a bijective function.

It's a cubic function, then its reach $ mathbb {R} $is therefore surjective.

Now I have to check if it is injective. This is: $ x_1 (x_1 ^ 2 – 1) = x_2 (x_2 ^ 2 -1) $If I can find an example where this is true and $ x_1 neq x_2 $ I can say that's not injective, but it took a long time for me to find a counterexample $ x_1 = 1 $and $ x_2 = -1 $,

Imagine that I have a more complex function

$ f (x) = frac {x ^ 7 + x ^ 3-x + 7} {x ^ 4} $

When we draw a simple function, you make a line parallel to the axis of the abscissa, and when it touches two or more points of the function it is determined that it is not injective.

But what about the features we can not easily plot graphically? What is the algebraic method to determine if a function is injective without laboriously looking for counter examples in functions that require a very long computation?

Of course, remember that I am still a student and have no knowledge of analysis. But if the only general method requires a calculus, I am open to learning it. Thank you in advance.

Algebra precalculus – What is the ratio of the sum of all prime numbers under N to the sum of all the composite ones under N?

Ratio of the sum of primes below N to the sum of the composition below N

The picture is the graph of the ratio of the sum of all primes below 2,000,000
Sum of all compound numbers below 2,000,000 that I created with Python and Matplotlib.

[X-axis = number range]

[Y-axis = ratio]

It seems that the ratio approaches 0 when we go to $ infty $,

Can someone explain what it implies? What does it say about the distribution of primes? What is the reason for such behaviors? Is it okay to conclude that the curve gets infinitely closer to 0 for larger values ​​without ever increasing?

Algebra Precalculus – Convert between Cartesian and polar equations and their graphs

I am asked to convert $ (x + 2) ^ 2 + y ^ 2 = 4 $ in a polar equation, and then confirm on my calculator. I converted it to $ r = 4cos ( theta) $but the graphics do not look the same at all. If someone could explain how to check for each answer and why the graphics do not look the same, that would be great.

Algebra Precalculus – A polynomial in disguise. What am I missing conceptually?

The equation family:

$$ e ^ { frac {s} { ln (x)}} = e ^ { frac {t} { ln (1-x)}} $$ $ s, t in Bbb N $

$ x in Bbb A (0,1) $

Is a family of polynomial equations in disguise. It can be rewritten as:

$ x ^ t = (1-x) ^ s. $

Draw the parameter space for all $ s, t $ is equivalent to drawing the null set of this family of polynomial equations in $ (0,1) ^ 2. $

On the one hand, to draw points that satisfy $ x ^ t = (1-x) ^ s, $ yields points of the form $ P ( Bbb A, Bbb A). $

On the other hand, points are plotted that satisfy $ e ^ { frac {s} { ln (x)}} = e ^ { frac {t} { ln (1-x)}}, $ yields points of the form $ P ( Bbb A, y) $

The $ y- $Coordinates can be numbers in $ Bbb Q, Bbb T, Bbb A. $

If these two representations are equivalent to the family of algebraic equations, then why can not I determine the nature of the algebraic equations? $ y- $Coordinate? What do I miss here? Are they in fact fundamentally different equations, but between them is a map in the form of algebraic manipulations?

Algebra Precalculus – Should the set of irrational numbers be subdivided into more subsets?

The real numbers are divided into two subcategories. the rational numbers, those that can be written as a relationship, and the irrational ones, those that can not.

But we know that many of the irrational numbers, though they can not be written as proportion or fraction, can still be written as continue Fraction. So my question is, are there any more subdivisions of irrational numbers into those that can be written as a regular continuous fraction (whose nominators and denominators follow a regular pattern instead of being random), and those that are or should be tilted? By "present" I mean the amount that has been assigned a specific letter and that is officially recognized as a set of numbers.

Algebra precalculus – Define the domain of $ f (x) = sqrt {x ^ 2-4} $

I'm supposed to be the area of $ f (x) = sqrt {x ^ 2-4} $,

I arrived $ (- 2, infty) $ while the textbook solution is $ (- infty infty) $,

To get to my solution, I set the radicand to greater than or equal to zero:

$ x ^ 2 + 4 ge0 $

$ x + 2 ge0 $ # Square root of each page

$ x ge-2 $

This will give me the domain as $ (- 2, infty) $,

Why is the domain actually $ (- infty infty) $?

Algebra precalculus – solving two equations in three variables

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