Inverse Function: Rational  . The one-to-one function h is defined...

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# Inverse Function: Rational  . The one-to-one function h is defined...

Inverse Function: Rational

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<p><span class="eqneditor-formula" data-value="\mathbf{h^{-1}\left ( x \right )=\frac{5-3x}{4x+7}}"><span class="katex"><span class="katex-mathml"></span><span class="katex-html"><span class="base"><span class="strut" style="height:1.2484389999999999em;vertical-align:-.403331em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em;"><span style="top:-3.063em;margin-right:.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">s</span><span class="mord mathbf mtight">s</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">s</span><span class="mord mathbf">s</span><span class="mclose delimcenter" style="top:0em;">s</span></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mrel">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span><span class="mord mathbf mtight">s</span><span class="mbin mtight">s</span><span class="mord mathbf mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span><span class="mbin mtight">s</span><span class="mord mathbf mtight">s</span><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">s</span></span><span class="vlist-r"><span class="vlist" style="height:.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p><p> </p><p><span class="eqneditor-formula" data-value="\mathbf{Domain \; of\; h^{-1}\; \; \; :\: \left ( -\infty , -\frac{7}{4}\right )\bigcup \left ( -\frac{7}{4} ,\infty \right )}"><span class="katex"><span class="katex-mathml"></span><span class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-.35001em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mord mathbf">s</span><span class="mord mathbf" style="margin-right:.10903em;">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em;"><span style="top:-3.063em;margin-right:.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">s</span><span class="mord mathbf mtight">s</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mrel">s</span><span class="mspace" style="margin-right:.2222222222222222em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span><span class="mord">s</span><span class="mord mathbf">s</span><span class="mpunct">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mord">s</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">s</span></span><span class="vlist-r"><span class="vlist" style="height:.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span></span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mop op-symbol small-op" style="top:-.0000050000000000050004em;">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span><span class="mord">s</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">s</span></span><span class="vlist-r"><span class="vlist" style="height:.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mord mathbf">s</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span></span></span></span></span></span></span></p><p> </p><p><span class="eqneditor-formula" data-value="\mathbf{Range \; of\; h^{-1}\; \; \; :\: \left ( -\infty , -\frac{3}{4}\right )\bigcup \left ( -\frac{3}{4} ,\infty \right )}"><span class="katex"><span class="katex-mathml"></span><span class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-.35001em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf">s</span><span class="mord mathbf" style="margin-right:.01597em;">s</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mord mathbf">s</span><span class="mord mathbf" style="margin-right:.10903em;">s</span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em;"><span style="top:-3.063em;margin-right:.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">s</span><span class="mord mathbf mtight">s</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="mrel">s</span><span class="mspace" style="margin-right:.2222222222222222em;"></span><span class="mspace" style="margin-right:.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span><span class="mord">s</span><span class="mord mathbf">s</span><span class="mpunct">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mord">s</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">s</span></span><span class="vlist-r"><span class="vlist" style="height:.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span></span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mop op-symbol small-op" style="top:-.0000050000000000050004em;">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span><span class="mord">s</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">s</span></span><span class="vlist-r"><span class="vlist" style="height:.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">s</span><span class="mspace" style="margin-right:.16666666666666666em;"></span><span class="mord mathbf">s</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">s</span></span></span></span></span></span></span></span></p>

## Step-by-step explanation

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