Is $g=\left\{\left(1,1\right),\left(2,3\right),\left(3,5\right),\left(4,7\right)\right\}$ a function? If $g$ is described by $g\left(x\right)=\alpha x+\beta$ , then what value should be assigned to $\alpha$ and $\beta$ .

<p>This is a Short Answer Type Question as classified in NCERT Exemplar</p><p><strong>Sol:</strong></p><p>Given that, g = { (1, 1), (2, 3), (3, 5), (4, 7)}.</p><p>Here, each element of domain has unique image. So, g is a function.</p><p>Now given that, g (x ) =&nbsp;<math><mrow><mi>α</mi></mrow></math>&nbsp;x +&nbsp;<math><mrow><mi>β</mi></mrow></math></p><p>For (1,1) g (1) =&nbsp;<math><mrow><mi>α</mi></mrow></math>&nbsp;+&nbsp;<math><mrow><mi>β</mi></mrow></math></p><p><math><mrow><mi>α</mi></mrow></math>&nbsp;+&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= 1… (i)</p><p>For (2,3)&nbsp; &nbsp;g (2) = 2 <math><mrow><mi>α</mi></mrow></math>&nbsp;+&nbsp;<math><mrow><mi>β</mi></mrow></math></p><p>2&nbsp;<math><mrow><mi>α</mi></mrow></math>&nbsp;+&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= 3 … (ii)</p><p>From Equations. (i) and (ii), </p><p>2 (1 -&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;) +&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= 3</p><p>2 - 2&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;+&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= 3</p><p>2 -&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= 3</p><p><math><mrow><mi>β</mi></mrow></math>&nbsp;= - 1</p><p>If&nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= - 1, then&nbsp;<math><mrow><mi>α</mi></mrow></math>&nbsp;= 2</p><p><math><mrow><mo>∴</mo></mrow></math>&nbsp;<math><mrow><mi>α</mi></mrow></math>&nbsp;= 2, &nbsp;<math><mrow><mi>β</mi></mrow></math>&nbsp;= - 1</p>

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