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

question 12

Multiple Choice

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.
$$f ( x ) = - \frac { 5 } { 7 } x - 3 , \quad g ( x ) = - \frac { 7 x + 21 } { 5 }$$

• A)
• \(\begin{array}{rlrl}
• f(g(x)) & =-\frac{5}{7}\left(\frac{7 x+21}{5}\right)-3 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \\
• & =\left(\frac{7 x+21}{7}\right)-3 & & =-\frac{(-5 x-21)+21}{5} \\
• & =(x+3)-3 & & =\frac{-5 x-21+21}{5} \\
• & =x+3-3 & & \\
• & =x & &= \frac{5 x}{5}
• \end{array}\)
• B)
• \(\begin{array}{rlrl}
• f(g(x)) & =-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-21 & g(f(x)) & =-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \\
• & =\left(\frac{35 x+21}{35}\right)-21 & & =-\frac{(-5 x-3)+21}{5} \\
• & =(x+21)-21 & & =\frac{5 x+3-21}{5} \\
• & =x+21-21 & & =\frac{5 x}{5} \\
• & =x & &=x
• \end{array}\)
• C)
• \(\begin{array}{l}
• \begin{array}{c}
• f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+3}{5}\right)-3 \\
• &=\left(\frac{35 x+3}{35}\right)-3 \\
• &=(x+3)-3 \\
• &=x+3-3 \\
• &=x
• \end{array}
• \begin{array}{c}
• g(f(x))=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5}\\
• =-\frac{(-5 x-3)+3}{5}\\
• =\frac{5 x+3-3}{5}\\
• =\frac{5 x}{5}\\
• =x
• \end{array}
• \end{array}\)
• D)
• \(\begin{array}{l}
• \begin{array}{lll}
• f(g(x)) &=-\frac{5}{7}\left(-\frac{7 x+21}{5}\right)-3 \\
• &=\left(\frac{7 x+21}{7}\right)-3 \\
• &=(x+3)-3 \\
• &=x+3-3 \\
• &=x
• \end{array}
• \begin{array}{lll}
• g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \\
• &=-\frac{(-5 x-21)+21}{5} \\
• &=\frac{5 x+21-21}{5} \\
• &=\quad \frac{5 x}{5} \\
• &=x
• \end{array}
• \end{array}\)
• E)
• \(\begin{array}{l}
• \begin{array}{lll}
• f(g(x)) &=-\frac{7}{5}\left(-\frac{5 x+15}{7}\right)-3 \\
• &=\left(\frac{5 x+15}{5}\right)-3 \\
• &=(x+3)-3 \\
• &=x+3-3 \\
• &=x
• \end{array}
• \begin{array}{lll}
• g(f(x)) &=-\frac{7\left(-\frac{5}{7} x-3\right)+21}{5} \\
• &=-\frac{(-5 x-3)+21}{35} \\
• &=\frac{5 x+3-21}{35} \\
• &=\quad \frac{35 x}{35} \\
• &=\quad x
• \end{array}
• \end{array}\)

Apply production functions to calculate optimal levels of inputs for profit maximization.

Determine the optimal mixture of multiple inputs based on their relative costs and production function.

Calculate the effect of changes in input and output prices on profit-maximizing input levels.

Analyze the impact of marginal productivities on input usage decisions.

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Definitions:

Notes Payable

A written agreement where the borrower promises to pay back a certain amount of money, under specific terms, to the lender at a future date.

Unamortized Discount

The portion of a bond's issue price that is below its face value and has not yet been amortized over the life of the bond.

Unamortized Premium

The portion of a bond premium that has not yet been amortized or gradually written off over the life of the bond.

Installment Note

A debt instrument that requires a series of periodic payments (installments) of principal and interest over a specific period until the debt is paid off.