<p>Consider the following statements<br>\(\mathrm{A}\): \(\quad \log_{10}{(10^x)} &gt; 0\)<br>\(\mathrm{B}\): \(\quad -0.5 \leq \cos{(0.5x)} \leq 0.5\)<br>\(\mathrm{C}\): \(\quad -\frac{\pi}{2} \leq \tan^{-1}{x} \leq \frac{\pi}{2}\)<span class="inline-marks">[6]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Determine which statements are true for all real numbers \(x\). Write your answers (yes or no) in the table below.</div><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 200px;"><col style="width: 200px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>Statements</p></th><th style="text-align: center; --cell-justify: center"><p>(a) Is the statement true for all real numbers \(x\)? (Yes/No)</p></th><th style="text-align: center; --cell-justify: center"><p>(b) If not true, example</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>\(\mathrm{A}\)</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(\mathrm{B}\)</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(\mathrm{C}\)</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></li><li><div class="list-item-paragraph">If a statement is not true for all \(x\), complete the last column by giving an example of one value of \(x\) for which the statement is false.</div></li></ol>

Consider the following statements
\(\mathrm{A}\): \(\quad \log_{10}{(10^x)} > 0\)
\(\mathrm{B}\): \(\quad -0.5 \leq \cos{(0.5x)} \leq 0.5\)
\(\mathrm{C}\): \(\quad -\frac{\pi}{2} \leq \tan^{-1}{x} \leq \frac{\pi}{2}\)

Determine which statements are true for all real numbers \(x\). Write your answers (yes or no) in the table below.

If a statement is not true for all \(x\), complete the last column by giving an example of one value of \(x\) for which the statement is false.

Exponential and Logarithmic Functions

International Baccalaureate Standard Level Mathematics