Verified Answer :

Panoyin
  1. \(h(x) = \sqrt[3]{x} - 3\)
        \(y = \sqrt[3]{x} - 3\)
        \(x = \sqrt[3]{y} - 3\)
        \(x + 3 = \sqrt[3]{y}\)
        \((x + 3)^{3} = y\)
        \((x + 3)(x + 3)(x + 3) = y\)
        \((x(x + 3) + 3(x + 3))(x + 3) = y\)
        \((x(x) + x(3) + 3(x) + 3(3))(x + 3) = y\)
        \((x^{2} + 3x + 3x + 9)(x + 3) = y\)
        \((x^{2} + 6x + 9)(x + 3) = y\)
        \(x^{2}(x + 3) + 6x(x + 3) + 9(x + 3) = y\)
        \(x^{2}(x) + x^{2}(3) + 6x(x) + 6x(3) + 9(x) + 9(3) = y\)
        \(x^{3} + 3x^{2} + 6x^{2} + 18x + 9x + 27 = y\)
        \(x^{3} + 9x^{2} + 27x + 27 = y\)
        \(x^{3} + 9x^{2} + 27x + 27 = h^{-1}(x)\)

    2. \(g(x) = \frac{1}{x} - 2x\)
        \(y = \frac{1}{x} - 2y\)
        \(x = \frac{1}{x} - 2y\)
        \(x - \frac{1}{x} = -2y\)
        \(\frac{x - \frac{1}{x}}{-2} = \frac{-2y}{-2}\)
        \(-\frac{1}{2}x + \frac{1}{2x} = y\)
        \(-\frac{1}{2}x + \frac{1}{2x} = g^{-1}(x)\)
       
    3. \(f(x) = 4x\)
        \(y = 4x\)
        \(x = 4y\)
        \(\frac{x}{4} = \frac{4y}{4}\)
        \(\frac{1}{4}x = y\)
        \(\frac{1}{4}x = f^{-1}(x)\)

    4. \(f(x) = -\frac{1}{5}x - 1\)
        \(y = -\frac{1}{5}x - 1\)
        \(x = -\frac{1}{5}y - 1\)
        \(x + 1 = -\frac{1}{5}y\)
        \(-5(x + 1) = -5(-\frac{1}{5}y)\)
        \(-5(x) - 5(1) = y\)
        \(-5x - 5 = y\)
        \(-5x - 5 = f^{-1}(x)\)

    5. \(g(x) = \frac{1}{x - 1}\)
        \(y = \frac{1}{x - 1}\)
        \(y(x - 1) = x - 1(\frac{1}{x - 1})\)
        \(y(x) - y(1) = -1\)
        \(xy - y = -1\)
        \(xy = y - 1\)
        \(\frac{xy}{y} = \frac{y - 1}{y}\)
        \(x = 1 - \frac{1}{y}\)
        \(y = 1 - \frac{1}{x}\)
        \(f^{-1}(x) = 1 - \frac{1}{x}\)

    6. \(f(x) = -2x^{3} + 1\)
        \(y = -2x^{3} + 1\)
        \(x = -2y^{3} + 1\)
        \(x - 1 = -2y^{3}\)
        \(\frac{x - 1}{-2} = \frac{-2y^{3}}{-2}\)
        \(-\frac{1}{2}x + \frac{1}{2} = y^{3}\)
        \(\sqrt[3]{-\frac{1}{2}x + \frac{1}{2}} = y\)
        \(\sqrt[3]{-\frac{1}{2}x + \frac{1}{2}} = f^{-1}(x)\)

    7. \(g(x) = \frac{-x - 5}{3}\)
        \(y = \frac{-x - 5}{3}\)
        \(y(3) = 3(\frac{-x - 5}{3})\)
        \(3y = -x - 5\)
        \(3x + 5 = -y\)
        \(\frac{3x + 5}{-1} = \frac{-y}{-1}\)
        \(-3x - 5 = y\)
        \(-3x - 5 = f^{-1}(x)\)

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