geno3141

If the problems is stated as:  ax² + bx + c  =  0,

the quadratic formula is:        x  =  [ -b ± √(b² - 4ac) ] / (2a)

When graphed, this is a hyperbola. The question is:  What y-value is impossible?

If you find the lim(x→∞) [3x - 7] / [x + 1], you will be able to answer the question.

Rewrite  [3x - 7] / [x + 1]  =  [3 - 7/x] / [1 + 1/x]   (by dividing all terms by x0

--->    lim(x→∞) [3 - 7/x] / [1 + 1/x]  =  3.

To show that 3 is impossible, try to solve the equation:  3  =  (3x - 7) / (x + 1)

     Multiply both sides by  x + 1     --->     3(x + 1)  =  3x - 7     --->     3x + 3  =  3x - 7

     --->  3  =  -7   Impossible!

Thus, the range is all numbers but 3.

For numbers smaller than -1, g(x) will have a negative number under the radical sign, will be undefinded for real numbers.

However, f(x), for values less than -1, both the numerator and the denominator will be negative, giving a positive answer, and will then be defined.

The functions will be equaivalent if x > -1; for x ≤ -1, f(x) will be defined but g(x) won't be.

E(r, r, 3)  =  625  =  r x r^3  =  r^4

                     625  =  r^4

Find the fourth root of both sides:

                    625^(1/4)  =  r

                           --->   r  =  5

All numbers can be substituted for x except 0, therefore:  -∞ < x < 0  or  0 < x < ∞

--->   {-∞, 0}  ∪  {0, ∞}     --->   (-∞, 0)  ∪  (0, ∞) 

f(x)  =  3x² - 4x

f(k - x)  =  3(k - x)² - 4(k - x)  = 3(k² -2kx + kx²) - 4k + 4x  =  3k² - 6kx + 3x² - 4k + 4x 

for f(x)  =  f(k - x)     ---     3x² - 4x  =   3k² - 6kx + 3k² - 4k + 4x 

The 3x²  on the left side matches the 3x² on the right side, so we can ignore them leaving:

 - 4x  =   3k² - 6kx - 4k + 4x 

On the right side: 3k² - 4k is a constant; there is no equaivalent on the left side, so 3k² - 4k  =  0

--->     k(3k - 4)  =  0       --->     So either k = 0  or 3k - 4  =  0   --->  k = 0  or  k  =  4/3

What we have left:  -4x  =  -6kx + 4x

                                -4x  =  (-6k + 4)x

The only way that this can be true is if  -4  =  -6k + 4, or  k = 4/3

Answer:  k  =  4/3

Use the calculator at this site.

First, make certain that you are in the correct mode (lower left-hand corner), either radian or degree.

Then press (2nd) key.

Then press (asin) key.      "asin"  is "arcsin"  which is "invsin"

Enter 3/7 and press (=).

The answer is approximately 25 degrees.

-2  ≤  3x - 5  ≤  7

When solving an equation, what you do to one side, you have to do to both sides.

With an extended inequality like this, what you do to one part, you have to do to all three parts.

The problem is to get the middle part to just be 'x'.

First:  add 5 to all three parts:

3  ≤  3x   ≤  12

Now:  divide all three parts by 3:

1  ≤  x   ≤  4

A quadratic formula, when graphed, becomes a parabola. It will have two values from every value of x except for the vertex.

So, the problem becomes:  find the vertex.

This can be done by graphing, by completing the square, or by using this formula:  x  =  -b / (2a).

Because you have the function in the form:  f(x)  =  ax² + bx + c   --->  a = 1, b = 4, c = -31

 --->    x  =  -b / (2a)   --->   x  =  -4 / (2·1)  =  -2

The x-value of the vertex occurs when  x = -2, now:   f(-2)  =  (-2)² + 4(-2) - 31  =  -35 

This value, -35, is the answer that you are looking for.

Yes, it is.