Set, Relations and FunctionsMTP Nov 19Question 1987 of 217
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If f(x)=x2−4x−2then\displaystyle f(x) = \frac{x^2-4}{x-2} thenf(2)$ is.

Options

A0
B2
C4
D1
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Correct Answer

✅ Option d — 1

All Options:

  • A0
  • B2
  • C4
  • D1

Detailed Solution & Explanation

Let's evaluate the function at x=2\displaystyle x = 2:
f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2}
We factor the numerator as a difference of squares:
x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2)<br>So,for\displaystyle <br>So, forx \neq 2,thefunctionsimplifiesto:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mfrac><mrow><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mostretchy="false">)</mo><mostretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mostretchy="false">)</mo></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotationencoding="application/x−tex">f(x)=(x−2)(x+2)x−2=x+2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.1963em;vertical−align:−0.7693em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.427em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.7693em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span></span></span></span></span><br>Bycontinuity,evaluatingat\displaystyle , the function simplifies to:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f(x) = \frac{(x - 2)(x + 2)}{x - 2} = x + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1963em;vertical-align:-0.7693em;"></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:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span><br>By continuity, evaluating atx = 2\displaystyle yields:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">f(2) = 2 + 2 = 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span><br>This corresponds to **Option C**.<br><br>**Discrepancy & Typographical Error:**<br>Mathematically, the value of the function (by standard algebraic simplification) is4(∗∗OptionC∗∗),whichiscorrectlykeyedinsimilarpapers(likeMTPMarch21).However,theanswerkeyforthisspecificmockpaper(MTPNov19)containsatypographicalerror,designating∗∗OptionD∗∗(1)asthecorrectanswer.Themathematicallycorrectresultisindeed\displaystyle (**Option C**), which is correctly keyed in similar papers (like MTP March 21). However, the answer key for this specific mock paper (MTP Nov 19) contains a typographical error, designating **Option D** (1) as the correct answer. The mathematically correct result is indeed4(OptionC).<br><br>Hence,∗∗OptionD∗∗isthecorrectanswer(accordingtothetextbookkey),butthemathematicallyprovencorrectvalueis\displaystyle (Option C).<br><br>Hence, **Option D** is the correct answer (according to the textbook key), but the mathematically proven correct value is4$ (**Option C**).

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