hectictar

If     f( -x )  =  f(x)     then the function is even.

If     f( -x )  =  -f(x)    then the function is odd.

\(f(x)\ =\ \frac{3}{2x^6-5}\)

                                           Plug in  -x  for  x

\(f(-x)\ =\ \frac{3}{2(-x)^6-5}\)

                                           And     (ab)^c  =  a^c b^c     so     (-x)⁶  =  (-1)⁶ x⁶

\(f(-x)\ =\ \frac{3}{2(-1)^6x^6\,-\,5}\)

                                           Because     -1 * -1  =  1,     -1  to an even power  =  1     and so     (-1)⁶  =  1

\(f(-x)\ =\ \frac{3}{2x^6\,-\,5}\)

                                           Now notice that  \(f(x)\ =\ \frac{3}{2x^6-5}\)  so we can substitute  f(x)  in for  \(\frac{3}{2x^6-5}\)

\(f(-x)\ =\ f(x)\)

Since  f(-x)  =  f(x) ,  the function is even.

\(y\ =\ \frac{x^3-x^2+x}{6x^2-9x}\)

Hmmm....even if you cut the pictures into 1 inch by 1 inch squares, there's just not enough square inches on the board to fit 70 pictures worth.

number of square inches on the board  =  32 * 32  =  1024

number of square inches of 70 pictures  =  70 * 3 * 5  =  1050

Are you sure it says  70  and not  60  ?

If so, maybe you can be the one to report the error in this question. Yes it is weird and unlikely they'd have it wrong, but it's not impossible.

If    (x + 3)²  ≤  1    then the absolute value of   x + 3   must be less than  1

| x + 3 |  ≤  1

Then in order for that to be true, it must be that

-1  ≤  x + 3  ≤  1

Subtract  3  from all parts of the inequality.

-4  ≤  x  ≤  -2

Then the integer solutions to that inequality are clearly

x = -4,  x = -3,  x = -2

If     f( -x )  =  f(x)     then the function is even.

If     f( -x )  =  -f(x)    then the function is odd.

\(f(x) = \dfrac{5^x - 1}{5^x + 1}\)

                                       Plug in  -x  for  x

\(f(-x) = \dfrac{5^{(-x)} - 1}{5^{(-x)} + 1}\)

                                       Now we are looking for a way rewrite the right side so that  f(x)  appears.

                                       Let's rewrite  5^(-x)  as  1 / 5^x

\(f(-x) = \dfrac{\frac{1}{5^x} - 1}{\frac{1}{5^x} + 1}\)

                                             Multiply the numerator and denominator by  5^x

\(f(-x) = \dfrac{\frac{1}{5^x} - 1}{\frac{1}{5^x} + 1}\cdot\dfrac{5^x}{5^x}\)

                                             Distribute the  5^x  to the terms in the numerator and denominator

\(f(-x) = \dfrac{1 - 5^x}{1 + 5^x}\)

                                             Factor  -1  out of the numerator.

\(f(-x) = \dfrac{-1(-1 + 5^x)}{1 + 5^x}\)

                                             Addition can be done in any order so we can rearrange the terms like this..

\(f(-x) = \dfrac{-1( 5^x-1)}{ 5^x+1}\)

                                             Now we can write the  -1  beside the fraction like this...

\(f(-x) = -1\cdot\dfrac{ 5^x-1}{ 5^x+1}\)

                                             Finally, f(x)  has appeared!  \(f(x) = \dfrac{5^x - 1}{5^x + 1}\)   so we can substitute  f(x)  in for  \(\dfrac{5^x - 1}{5^x + 1}\)

\(f(-x) = -1\cdot f(x)\)

\(f(-x) = - f(x)\)

Remember that if     f( -x )  =  -f(x)    then the function is odd.

Since  f( -x )  =  -f(x)  , the function is odd.

As Rom said, there is definitely no way to put 70 pictures on the board.

68 pictures is the maximum possible, and here is a way to do that:

x² + 5x  <  6

                            Subtract  6  from both sides of the inequality.

x² + 5x - 6  <  0

                            Let's find what values of  x  make  x² + 5x - 6  equal  0

x² + 5x - 6  =  0

                            Factor the left side. What two numbers add to  5  and multiply to  -6 ?   -1  and  +6

(x - 1)(x + 6) = 0

                            Set each factor equal to zero and solve for  x

Did you mean:

(ZY)² + 2(WY)(WZ) cos(38°)  =  (XY)² + 2(WY)(WZ) cos(41°)

and     cos(41°)  <  cos(38°)

so       | ZY |  <  | XY |

????

A conditional statement is a statement that can be expressed as "If _____, then _____."

The statement  "I will eat anything that is dipped in chocolate,"  can be expressed as  "If something is dipped in chocolate, then I will eat it."