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)
• \(\begin{aligned}
• f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} & g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\
• &=\sqrt[3]{\left(x^{3}+\frac{7}{8}\right)-7} & & =\frac{24 x-21+21}{24} \\
• &=\sqrt[3]{x^{3}+\frac{0}{8}} & & =\frac{24 x}{24} \\
• &=\sqrt[3]{x^{3}} & & =x \\
• &=x
• \end{aligned}\)

---

Definitions:

Income Before Taxes

Another term for pretax income, indicating the amount earned by a business before subtracting taxes.

Loss On Sale

Loss on sale occurs when the selling price of an asset is less than its carrying amount or book value, resulting in a financial loss for the seller.