# Test: Functions I - Ambitious

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Question 1:   Which of the functions is one-to-one?
$f\left(x\right)={x}^{2},\left\{x\in ℝ\right\}$
$f\left(x\right)=\mathrm{cos}x,\left\{x:-\frac{\pi }{2}\le x\le \frac{\pi }{2}\right\}$
$f\left(x\right)={x}^{2},\left\{x:0\le x\le +\infty \right\}$
$f\left(x\right)=\mathrm{sin}x,\left\{x:-\pi \le x\le \pi \right\}$
Question 2:   Which of the functions is many-to-one?
$f\left(x\right)=\mathrm{sin}x,\left\{x:\frac{3}{2}\pi
$f\left(x\right)=\mathrm{tan}x,\left\{x:0\le x\le \pi \right\}$
$f\left(x\right)=\mathrm{sin}x,\left\{x:\frac{\pi }{2}\le x\le \frac{3}{2}\pi \right\}$
$f\left(x\right)=\mathrm{tan}x,\left\{x:\pi
Question 3:   Find the inverse of the function $f\left(x\right)={x}^{2}-3,x\ge 0$
${f}^{-1}\left(x\right)=\sqrt{x}+3$
${f}^{-1}\left(x\right)=\sqrt{x+3}$
${f}^{-1}\left(x\right)=\frac{3}{x}$
${f}^{-1}\left(x\right)=\frac{1}{x}+3$
Question 4:   Find the inverse of the function $f\left(x\right)={0.3}^{x}$
${f}^{-1}\left(x\right)=\frac{0.3}{x}$
${f}^{-1}\left(x\right)=0.3\mathrm{log}x$
${f}^{-1}\left(x\right)={\mathrm{log}}_{0.3}x$
${f}^{-1}\left(x\right)={x}^{0.3}$
Question 5:   Find $g\left(f\left(3\right)\right)$, where $f\left(x\right)=\frac{1}{3}{x}^{2},g\left(x\right)=2x+3$
$g\left(f\left(3\right)\right)=6$
$g\left(f\left(3\right)\right)=9$
$g\left(f\left(3\right)\right)=27$
$g\left(f\left(3\right)\right)=15$
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