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

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Definitions:

Direct Lease

A leasing agreement where the lessor buys the asset chosen by the lessee and then leases it back to them.

Lessor

The party that leases or rents out a property or asset to another party, known as the lessee.

Present Value

The present value of a future amount of money or series of cash inflows, discounted by a particular return rate.

Fair Market Value

The price at which an asset would trade in a competitive auction setting, reflecting the value agreed upon by a willing buyer and seller.