Answer :
To determine which pair of functions are inverses of each other, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \) for each option.
### Option A: \( f(x) = \frac{x}{7} + 10 \) and \( g(x) = 7x - 10 \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f(7x - 10) = \frac{7x - 10}{7} + 10 = x - \frac{10}{7} + 10 = x + 10 - \frac{10}{7} \][/tex]
Since \( 10 - \frac{10}{7} \neq 0 \), \( f(g(x)) \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{x}{7} + 10\right) = 7\left(\frac{x}{7} + 10\right) - 10 = x + 70 - 10 = x + 60 \][/tex]
Again, \( x + 60 \neq x \).
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option A.
### Option B: \( f(x) = \frac{7}{x} - 2 \) and \( g(x) = \frac{x + 2}{7} \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x + 2}{7}\right) = \frac{7}{\frac{x + 2}{7}} - 2 = \frac{7 \cdot 7}{x + 2} - 2 = \frac{49}{x + 2} - 2 \][/tex]
Clearly, \( \frac{49}{x + 2} - 2 \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{7}{x} - 2\right) = \frac{\frac{7}{x} - 2 + 2}{7} = \frac{7/x}{7} = \frac{1}{x} \][/tex]
Clearly, \( \frac{1}{x} \neq x \).
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option B.
### Option C: \( f(x) = \sqrt[3]{11x} \) and \( g(x) = \left(\frac{x}{11}\right)^3 \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\left(\frac{x}{11}\right)^3\right) = \sqrt[3]{11 \left(\frac{x}{11}\right)^3} = \sqrt[3]{\frac{11 x^3}{11^3}} = \sqrt[3]{\frac{x^3}{11^2}} = \frac{x}{11} \][/tex]
Clearly, \( \frac{x}{11} \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g(\sqrt[3]{11x}) = \left(\frac{\sqrt[3]{11x}}{11}\right)^3 = \left(\frac{x}{11}\right)^3 \][/tex]
Clearly, this simplification shows the inconsistency.
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option C.
### Option D: \( f(x) = 9x - 6 \) and \( g(x) = \frac{x + 6}{9} \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x + 6}{9}\right) = 9 \left(\frac{x + 6}{9}\right) - 6 = x + 6 - 6 = x \][/tex]
\( f(g(x)) = x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g(9x - 6) = \frac{(9x - 6) + 6}{9} = \frac{9x}{9} = x \][/tex]
\( g(f(x)) = x \).
Thus, \( f(x) \) and \( g(x) \) are inverses of each other in Option D.
The pair of functions that are inverses of each other is given in Option D: [tex]\( f(x) = 9x - 6 \)[/tex] and [tex]\( g(x) = \frac{x + 6}{9} \)[/tex].
### Option A: \( f(x) = \frac{x}{7} + 10 \) and \( g(x) = 7x - 10 \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f(7x - 10) = \frac{7x - 10}{7} + 10 = x - \frac{10}{7} + 10 = x + 10 - \frac{10}{7} \][/tex]
Since \( 10 - \frac{10}{7} \neq 0 \), \( f(g(x)) \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{x}{7} + 10\right) = 7\left(\frac{x}{7} + 10\right) - 10 = x + 70 - 10 = x + 60 \][/tex]
Again, \( x + 60 \neq x \).
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option A.
### Option B: \( f(x) = \frac{7}{x} - 2 \) and \( g(x) = \frac{x + 2}{7} \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x + 2}{7}\right) = \frac{7}{\frac{x + 2}{7}} - 2 = \frac{7 \cdot 7}{x + 2} - 2 = \frac{49}{x + 2} - 2 \][/tex]
Clearly, \( \frac{49}{x + 2} - 2 \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{7}{x} - 2\right) = \frac{\frac{7}{x} - 2 + 2}{7} = \frac{7/x}{7} = \frac{1}{x} \][/tex]
Clearly, \( \frac{1}{x} \neq x \).
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option B.
### Option C: \( f(x) = \sqrt[3]{11x} \) and \( g(x) = \left(\frac{x}{11}\right)^3 \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\left(\frac{x}{11}\right)^3\right) = \sqrt[3]{11 \left(\frac{x}{11}\right)^3} = \sqrt[3]{\frac{11 x^3}{11^3}} = \sqrt[3]{\frac{x^3}{11^2}} = \frac{x}{11} \][/tex]
Clearly, \( \frac{x}{11} \neq x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g(\sqrt[3]{11x}) = \left(\frac{\sqrt[3]{11x}}{11}\right)^3 = \left(\frac{x}{11}\right)^3 \][/tex]
Clearly, this simplification shows the inconsistency.
Thus, \( f(x) \) and \( g(x) \) are not inverses in Option C.
### Option D: \( f(x) = 9x - 6 \) and \( g(x) = \frac{x + 6}{9} \)
1. Calculate \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x + 6}{9}\right) = 9 \left(\frac{x + 6}{9}\right) - 6 = x + 6 - 6 = x \][/tex]
\( f(g(x)) = x \).
2. Calculate \( g(f(x)) \):
[tex]\[ g(f(x)) = g(9x - 6) = \frac{(9x - 6) + 6}{9} = \frac{9x}{9} = x \][/tex]
\( g(f(x)) = x \).
Thus, \( f(x) \) and \( g(x) \) are inverses of each other in Option D.
The pair of functions that are inverses of each other is given in Option D: [tex]\( f(x) = 9x - 6 \)[/tex] and [tex]\( g(x) = \frac{x + 6}{9} \)[/tex].
