<PRE><B>Question 185447</B>
There&#39;s a problem here: the equation {{{y=2x^2-3}}} has no inverse over the domain *[Tex \LARGE \left(-\infty,\infty\right)]. However, if we restrict the domain to be &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE 0,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;, then we can find the inverse.



So...

Domain: &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE 0,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt; 

Range: &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE -3,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;



{{{y=2x^2-3}}} Start with the given equation. 



{{{x=2y^2-3}}} Switch each &quot;x&quot; and &quot;y&quot;. The goal is to now solve for &quot;y&quot;



{{{x+3=2y^2}}} Add 3 to both sides.



{{{(x+3)/2=y^2}}} Divide both sides by 2.



{{{y^2=(x+3)/2}}} Rearrange the equation



{{{y=sqrt((x+3)/2)}}} Take the square root of both sides. Note: since we made the domain &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE 0,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;, this means that we&#39;re only dealing with the positive square root.



So the inverse equation is {{{y=sqrt((x+3)/2)}}} where the domain and range is simply flipped:


Domain: &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE -3,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;

Range: &lt;font size=&quot;6&quot;&gt;[&lt;/font&gt;*[Tex \LARGE 0,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;</PRE>