When the definition of a function involves a fraction, the function is undefined at any value that would make the denominator of the function

zero

Step-by-step explanation:

Given a rational function then the denominator cannot be zero as this would make the function undefined.

zero

Step-by-step explanation:

We have to fill the correct word in the blank space.

Suppose

Substitute x=0 then we get

The function is not defined at x=0

Because the denominator of the function at x=0 is zero which makes the function not define.

When the definition of a function involves  a fraction , the function is undefined at any value that would make the denominator of the function zero.

 The correct answer is:  " 6 " .

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         { that is;  the simplified expression given is:  " 6" [positive 6].

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Step-by-step explanation:

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Given the expression;  which is intended to be:

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    Simplify:     " 1 + √ [ - (-5²)( - 1³) ] " ?

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Note:

Refer to "comments below" typed directly under 'where the question itself is asked' —for clarification.

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First, solve and simplify the value under the square root sign:

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- (-5²) (-1 ³) =  ?

   → (- 1)( - 5²) (- 1³)  = ?

    ↔  (- 5²)(- 1)( - 1³ )  = ?

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    - 5²  =  -5 * -5  = 25 ;  {Note:  A negative * a negative = a positive ).  

     →  Now, multiply this value by:  "(-1)" ;  

     (-1) * (25) =  -25 ;  

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Now,   (-25) * (-1³)  = ?

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Let us simplify:  " ( - 1 ³ ) " ;  

             " ( - 1³ )  =  (-1 * -1 * -1)  =  

                          ( -1 * -1) * -1  = ?  ;  

Note:  " (-1 * -1) =  1 (positive one) ;  

                   →  {since:  " a negative * a negative = a positive " .}.

               Then,  1 * (-1) =   -1  (negative one);  

      →  {since:  a "positive times a negative =  a negative"} ;  

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So:   " ( - 1³)   =  - 1  " ;

So:  (-25) * (-1³)  =  (-25) * (-1) = 25 (positive 25) ;  

→  {since:  a "negative times a negative = a positive" ; and since:

" any value, multiplied by "1" ; results in that same original value."}.  

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So, back to our problem:  

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Simplify:     " 1  +  √ [ - ( - 5² ) ( - 1 ³ ) " ?

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                               =    1   +  √25 ;  _______________________________________________

{Note that:  "√25 = 5 " ;   25 is a "perfect square";  

  since:  5² = 5 * 5 = 25 ;  and "25" is known as a "perfect square" ; since the square root of the non-zero positive integer  [in this case:  " 25 ") ;   results in a non-zero positive integer [in this case:  " 5 " .].

If not known by trial and error; or by common known knowledge, one may use a  calculator to find:  " √25 " ;  to get:  " 5 " .

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 So:   We can write our original expression:

            "  1  +   " ;

                     =  1   +  5  ;

                     =   6 .

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The correct answer is:  " 6 " {positive 6}.

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Hope this answer — and explanation — is helpful to you!

      Wishing you the best in your academic endeavors

                    — and within the "" community!

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