1. Sketch the graph of $\displaystyle y=5^{\left\vert x\right\vert }$

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  2. Solve the equation $\displaystyle 125^{x}\,5^{x+1} = ( {{1}\over{5}}) 5^{2\,x}$

    [ x = −1 ]

    [ x = 2 ]

    $\displaystyle \left[ x=-{{2}\over{5}} \right] $

    $\displaystyle \left[ x=-{{1}\over{155}} \right] $

  3. Let $f(x) = \left\vert x\right\vert $ , and let $g(x) = \sqrt{x-2}-1$ . Sketch the graph of  ${f\circ g}$.

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  4. If a linear function $f$ satisfies f(4) = 1 and f(6) = −2, find $f(x)$.

    $\displaystyle y={{x}\over{4}}-{{7}\over{2}}$

    $\displaystyle y=-{{3\,x}\over{2}}-7$

    $\displaystyle y=7-{{3\,x}\over{2}}$

    $\displaystyle y={{x}\over{4}}+{{7}\over{2}}$

  5. Solve the equation $\displaystyle \ln x=\ln \left(x+2\right)-1$

    $\displaystyle \left[ x={{2}\over{e-1}} \right] $

    [ x = 1 ]

    [ x = 2 ]

    $\displaystyle \left[ x=-{{2\,e}\over{e-1}} \right] $

  6. Find the vertex of $\displaystyle -x^2-4\,x-3$.

    [ 2 ,  1 ]

    [ 2 ,   − 1 ]

    [  − 2 ,   − 1 ]

    [  − 2 ,  1 ]

  7. Solve the equation $\displaystyle x^2\,e^{x}-2\,x\,e^{x}=0$

    $\displaystyle \left[ x=e , x=e^2 \right] $

    $\displaystyle \left[ x=0 , x=e^2 \right] $

    $\displaystyle \left[ x=-\sqrt{2} , x=\sqrt{2} \right] $

    [ x = 0 ,  x = 2 ]

  8. Solve the equation $\displaystyle 2\,x^2+4\,x+1=0$.

    $\displaystyle \left[ x=-{{\sqrt{2}+2}\over{2}} , x={{\sqrt{2}-2}\over{2}} \right] $

    $\displaystyle \left[ x=-\sqrt{3}-2 , x=\sqrt{3}-2 \right] $

    $\displaystyle \left[ x=-{{\sqrt{2}-2}\over{2}} , x={{\sqrt{2}+2}\over{2}} \right] $

    $\displaystyle \left[ x=-{{\sqrt{6}+2}\over{2}} , x={{\sqrt{6}-2}\over{2}} \right] $

  9. Solve the equation $\displaystyle 4^{x}=3^{2-x}$

    $\displaystyle \left[ x={{2}\over{\ln 4+\ln 3}} \right] $

    [ x = 1 ]

    $\displaystyle \left[ x={{2\,\ln 3}\over{\ln 4+\ln 3}} \right] $

    [ x = 2 ]

  10. Let $f(x) = \sqrt{x}-2$ , and let $g(x) = 9-x^2$ . Sketch the graph of  ${f\circ g}$.

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Department of Mathematics
Last modified: 2026-04-20