@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-NG5DMZ5W-1
  skos:prefLabel "fonction hyperbolique réciproque"@fr, "inverse hyperbolic function"@en ;
  a skos:Concept ;
  skos:broader psr:-WZWTRVZJ-X .

psr: a skos:ConceptScheme .
psr:-ZF39Q858-W
  skos:prefLabel "inverse trigonometric function"@en, "fonction trigonométrique inverse"@fr ;
  a skos:Concept ;
  skos:broader psr:-WZWTRVZJ-X .

psr:-WZWTRVZJ-X
  dc:modified "2023-07-26"^^xsd:date ;
  skos:narrower psr:-ZF39Q858-W, psr:-NG5DMZ5W-1 ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:altLabel "réciproque"@fr, "bijection réciproque"@fr ;
  skos:prefLabel "fonction réciproque"@fr, "inverse function"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Inverse_function>, <https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque> ;
  skos:broader psr:-L2BN0W1T-P, psr:-T88XBMNP-M ;
  skos:definition """En mathématiques, la <b>bijection réciproque</b> (ou <b>fonction réciproque</b> ou <b>réciproque</b>) d'une bijection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> est l'application qui associe à chaque élément de l'ensemble d'arrivée son unique antécédent par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span>. Elle se note <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f^{-1}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="f^{-1}"></span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque">https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque</a>)"""@fr, """In mathematics, the <b>inverse function</b> of a function <span class="texhtml mvar" style="font-style:italic;">f</span> (also called the <b>inverse</b> of <span class="texhtml mvar" style="font-style:italic;">f</span>) is a function that undoes the operation of <span class="texhtml mvar" style="font-style:italic;">f</span>. The inverse of <span class="texhtml mvar" style="font-style:italic;">f</span> exists if and only if <span class="texhtml mvar" style="font-style:italic;">f</span> is bijective, and if it exists, is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f^{-1}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d2b575826d75cfdbc1bea2f34ccfa71f1c59b7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.3ex; height:3.009ex;" alt="{\\\\displaystyle f^{-1}.}"></span>
<br/>For a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f\\\\colon X\\	o Y}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>:<!-- : --></mo>
<br/>        <mi>X</mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mi>Y</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\colon X\\	o Y}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="f\\\\colon X\\	o Y"></span>, its inverse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f^{-1}\\\\colon Y\\	o X}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>:<!-- : --></mo>
<br/>        <mi>Y</mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mi>X</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}\\\\colon Y\\	o X}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a1bc0edf199414feff53da55c19b265bc5015a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:3.009ex;" alt="{\\\\displaystyle f^{-1}\\\\colon Y\\	o X}"></span> admits an explicit description:  it sends each element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle y\\\\in Y}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>y</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>Y</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle y\\\\in Y}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="y\\\\in Y"></span> to the unique element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x\\\\in X}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>X</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\in X}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="x\\\\in X"></span> such that <span class="texhtml"><i>f</i>(<i>x</i>) = <i>y</i></span>.
<br/>As an example, consider the real-valued function of a real variable given by <span class="texhtml"><i>f</i>(<i>x</i>) = 5<i>x</i> − 7</span>. One can think of <span class="texhtml mvar" style="font-style:italic;">f</span> as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of <span class="texhtml mvar" style="font-style:italic;">f</span> is the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f^{-1}\\\\colon \\\\mathbb {R} \\	o \\\\mathbb {R} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>:<!-- : --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}\\\\colon \\\\mathbb {R} \\	o \\\\mathbb {R} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fe95e7e82c9eea2cd86f6b9789fd811a56bac4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:11.657ex; height:3.009ex;" alt="{\\\\displaystyle f^{-1}\\\\colon \\\\mathbb {R} \\	o \\\\mathbb {R} }"></span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f^{-1}(y)={\\rac {y+7}{5}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>y</mi>
<br/>              <mo>+</mo>
<br/>              <mn>7</mn>
<br/>            </mrow>
<br/>            <mn>5</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}(y)={\\rac {y+7}{5}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fee2ee9786aa2e2f5f84330f6a71e297ab6a087" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:16.358ex; height:5.343ex;" alt="{\\\\displaystyle f^{-1}(y)={\\rac {y+7}{5}}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Inverse_function">https://en.wikipedia.org/wiki/Inverse_function</a>)"""@en ;
  dc:created "2023-07-26"^^xsd:date .

psr:-T88XBMNP-M
  skos:prefLabel "set theory"@en, "théorie des ensembles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-WZWTRVZJ-X .

psr:-L2BN0W1T-P
  skos:prefLabel "fonction"@fr, "function"@en ;
  a skos:Concept ;
  skos:narrower psr:-WZWTRVZJ-X .