Given that,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmr10">(</span></span><span class="icmr10">1)</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Let, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span></span></nobr></span> then,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(0</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(0</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">k</span><span class="spacer" style="margin-left:0.031em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="icmr10">[</span>Let, f(0)=k<span class="icmr10">]</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Now proving <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> is onto function,
proof: <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">k</span><span class="spacer" style="margin-left:0.031em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
For all <span class="typeset"><nobr><span class="scale"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span>, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span>
Range <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span> <span class="typeset"><nobr><span class="scale"><span class="icmr10">=</span></span></nobr></span> codomain
<span class="typeset"><nobr><span class="scale"><img src="
jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span></span></nobr></span> is onto function
<span class="typeset"><nobr><span class="scale"><img src="
jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span></span></nobr></span> is a onto function Proof is in link(
https://www.youtube.com/channel/UCyzsvF ... MRQ/videos )
This video proves if <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">))</span></span></nobr></span> is onto so is <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> , we can use it in my proof by replacing <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> by <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span></nobr></span>.
Now back to the main solution,
Let, for some <span class="typeset"><nobr><span class="scale"><span class="icmmi10">t</span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span></span></nobr></span>, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">t</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">x</span></span></span></nobr></span> in (1) we get,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmr10">(</span></span><span class="icmr10">2)</span></span></nobr></span>
Finally , Let, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span></span></nobr></span> for some <span class="typeset"><nobr><span class="scale"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> which is real
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> [Using (2)]
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> [<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> using (1)]
Or <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char06.png" style=" width:11px; margin-right:-0.013em;"></span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span>
<span class="typeset"><nobr><span class="scale"><img src="
jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"></span></nobr></span> the required function <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char06.png" style=" width:11px; margin-right:-0.013em;"></span><span class="icmmi10">x</span></span></nobr></span>