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27.1: Review of functions and graphs

  • Page ID
    68482
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    Exercise \(\PageIndex{1}\)

    Find all solutions of the equation: \(\big| 3x-9\big| =6\)

    Answer

    \(x = 1\) or \(x = 5\)

    Exercise \(\PageIndex{2}\)

    Find the equation of the line displayed below.

    clipboard_e6193cbb38730a86463aff4c9f1015643.png

    Answer

    \(y=-\dfrac{1}{2} x+1\)

    Exercise \(\PageIndex{3}\)

    Find the solution of the equation: \(x^3-3x^2+2x-2=0\) Approximate your answer to the nearest hundredth.

    Answer

    \(x \approx 2.52\)

    Exercise \(\PageIndex{4}\)

    Let \(f(x)=x^2-2x+5\). Simplify the difference quotient \(\dfrac{f(x+h)-f(x)}{h}\) as much as possible.

    Answer

    \(2 x-2+2 h\)

    Exercise \(\PageIndex{5}\)

    Consider the following graph of a function \(f\).

    clipboard_ebbf8b3d341de54a74f708abe18207254.png

    Find: domain of \(f\), range of \(f\), \(f(3)\), \(f(5)\), \(f(7)\), \(f(9)\).

    Answer

    domain \(D=[2,7]\), range \(R=(1,4], f(3)=2, f(5)=2, f(7)=4, f(9) \) is undefined

    Exercise \(\PageIndex{6}\)

    Find the formula of the graph displayed below.

    clipboard_e79aaccb6dd8116b811f41f51239d7955.png

    Answer

    \(f(x)=-x^{2}+2\)

    Exercise \(\PageIndex{7}\)

    Let \(f(x)=5x+4\) and \(g(x)=x^2+8x+7\). Find the quotient \(\left(\dfrac f g\right)(x)\) and state its domain.

    Answer

    \(\left(\dfrac{f}{g}\right)(x)=\dfrac{5 x+4}{x^{2}+8 x+7}=\dfrac{5 x+4}{(x+7)(x+1)}\) has domain \(D=\mathbb{R}-\{-7,-1\}\)

    Exercise \(\PageIndex{8}\)

    Let \(f(x)=x^2+\sqrt{x-3}\) and \(g(x)=2x-3\). Find the composition \((f\circ g)(x)\) and state its domain.

    Answer

    \((f \circ g)(x)=4 x^{2}-12 x+9+\sqrt{2 x-6}\) has domain \(D=[3, \infty)\)

    Exercise \(\PageIndex{9}\)

    Consider the assignments for \(f\) and \(g\) given by the table below.

    \[\begin{array}{|c||c|c|c|c|c|}
    \hline x & 2 & 3 & 4 & 5 & 6 \\
    \hline f(x) & 5 & 0 & 2 & 4 & 2 \\
    \hline g(x) & 6 & 2 & 3 & 4 & 1 \\
    \hline
    \end{array} \nonumber \]

    Is \(f\) a function? Is \(g\) a function? Write the composed assignment for \((f\circ g)(x)\) as a table.

    Answer

    \(f\) and \(g\) are both functions, and the composition is given by the table:

    \(\begin{array}{|c||c|c|c|c|c|}
    \hline x & 2 & 3 & 4 & 5 & 6 \\
    \hline \hline f(x) & 5 & 0 & 2 & 4 & 2 \\
    \hline g(x) & 6 & 2 & 3 & 4 & 1 \\
    \hline(f \circ g)(x) & 2 & 5 & 0 & 2 & \text { undef. } \\
    \hline
    \end{array}\)

    Exercise \(\PageIndex{10}\)

    Find the inverse of the function \(f(x)=\dfrac{1}{2x+5}\).

    Answer

    \(f^{-1}(x)=\dfrac{1}{2 x}-\dfrac{5}{2}\)


    This page titled 27.1: Review of functions and graphs is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.