Question 991271
.
Hello,
your question is incorrect.


The correct question is:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the domain and range of the quadratic <U>function</U> &nbsp;&nbsp;f(x)=x^2-8x-9.


An equation has no domain. &nbsp;The term &nbsp;"domain" is not defined for equations.  

It is defined for functions.

It is not applicable for equations. &nbsp;It is applicable for functions.


The same is for range. &nbsp;It is for functions, &nbsp;not for equations.


OK. &nbsp;Now I am ready to answer this question:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the domain and range of the quadratic function &nbsp;f(x)=x^2-8x-9.


1) &nbsp;The domain of a quadratic function is the entire number line, &nbsp;i.e &nbsp;the set of all real numbers. 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is true for any quadratic function. &nbsp;Particularly, &nbsp;it is true for the given function.


2) &nbsp;The range of a quadratic function is stretched from its minimal value to the positive infinity, &nbsp;if the parabola is &nbsp;U-shaped &nbsp;(has positive coefficient at {{{x^2}}}, 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;as it is in your case). 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the parabola is bottom-up &nbsp;(the quadratic function has negative coefficient at &nbsp;{{{x^2}}}), &nbsp;then its range is stretched from its maximal value to the negative infinity.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;in your case we need to find the minimal value of the quadratic function. &nbsp;For a quadratic function &nbsp;f(x) = &nbsp;{{{ax^2 + bx + c}}} &nbsp;the minimum is reached at &nbsp;x = -{{{b/(2a)}}} 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and is equal to f(-{{{(b/(2a))}}}). &nbsp;In your case &nbsp;&nbsp;-{{{b/(2a)}}} = {{{8/2}}} = 4 &nbsp;and &nbsp;f(4) = {{{4^2 -8*4 - 9}}} = 16 - 32 - 9 = -25.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;the range of your function is semi-infinite segment &nbsp;&nbsp;[-25, {{{infinity}}}). 

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 500, -3.5, 11.5, -30.5, 5.5,
          x^2 - 8x - 9 
)}}}

  </TD>
  </TR>
</TABLE>