Show Algebraically That the Functions $f$ And $g$ Shown Below Are Inverse Functions

question 53

Multiple Choice

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.
$$f ( x ) = \sqrt [ 3 ] { 8 x - 7 } , \quad g ( x ) = \frac { x ^ { 3 } + 7 } { 8 }$$

• A)
• \(\begin{aligned}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} & g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\sqrt[3]{\left(x^{3}+56\right)-56} & & \frac{8 x-7^{3}+7^{3}}{8} \\
• &=\sqrt[3]{x^{3}+56-56} & &=\frac{8 x}{8} \\
• &=\sqrt[3]{x^{3}} & &\\
• & =x&=x\\
• \end{aligned}\)
• B)
• \(\begin{aligned}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} & g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\sqrt[3]{\left(x^{3}+7\right)-7} & &=\frac{8 x-7+7}{8} \\
• &=\sqrt[3]{x^{3}+7-7} & &=\frac{8 x}{8} \\
• &=\sqrt[3]{x^{3}} & \\
• &=x&=x\\
• \end{aligned}\)
• C)
• \(\begin{array}{l}
• \begin{array}{lll}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\
• &=\sqrt[3]{\left(\frac{8 x^{3}+7}{8}\right)-7} \\
• &=\sqrt[3]{x^{3}+7-7} \\
• &=\sqrt[3]{x^{3}} \\
• &=x
• \end{array}
• \quad \quad
• \begin{array}{lll}
• g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\frac{8^{3} x-7+7}{8^{3}} \\
• &=\frac{8^{3} x}{8^{3}} \\
• \\
• &=x
• \end{array}
• \end{array}\)
• D)
• \(\begin{aligned}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} & g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\sqrt[3]{\left(8 x^{3}+56\right)-56} & &=\frac{8^{3} x-7^{3}+7^{3}}{8^{3}} \\
• &=\sqrt[3]{8 x^{3}+56-56} & & =\frac{8^{3} x}{8^{3}} \\
• &=\sqrt[3]{8 x^{3}} & & \\
• &=x & &=x
• \end{aligned}\)
• E)
• \(\begin{aligned}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} & g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\sqrt[3]{\left(x^{3}+\frac{7}{8}\right)-7} & & =\frac{24 x-21+21}{24} \\
• &=\sqrt[3]{x^{3}+\frac{0}{8}} & & =\frac{24 x}{24} \\
• &=\sqrt[3]{x^{3}} & & =x \\
• &=x
• \end{aligned}\)

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Definitions:

Boiling Point

The temperature at which a substance transitions from the liquid phase to the gas phase under atmospheric pressure.

Cis-

A prefix that indicates substituents are located on the same side of a double bond or ring structure.

Trans-hex-3-ene

A geometric isomer of hex-3-ene with the two substituents opposite to each other across the double bond, affecting its chemical properties and reactivity.

Major Organic Product

The primary product of an organic reaction, usually the one in greater yield when multiple products are possible.