To determine which pairs of functions are inverses of each other through function composition, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \) for each pair.
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First pair:
\( f(x) = 3x + 79 \)
\( g(x) = 3x - 79 \)- \( f(g(x)) = f(3x - 79) = 3(3x - 79) + 79 = 9x - 237 + 79 = 9x - 158 \) (not equal to \( x \))
- \( g(f(x)) = g(3x + 79) = 3(3x + 79) - 79 = 9x + 237 - 79 = 9x + 158 \) (not equal to \( x \))
Not inverses.
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Second pair:
\( f(x) = \frac{3x + 7}{9} \)
\( g(x) = \frac{3x - 7}{9} \)- \( f(g(x)) = f\left(\frac{3x - 7}{9}\right) = \frac{3\left(\frac{3x - 7}{9}\right) + 7}{9} = \frac{\frac{9x - 21 + 63}{9}}{9} = \frac{9x + 42}{81} = x \) (this is incorrect)
- \( g(f(x)) = g\left(\frac{3x + 7}{9}\right) = \frac{3\left(\frac{3x + 7}{9}\right) - 7}{9} = \frac{9x + 21 - 63}{81} = x \) (this is incorrect)
Not inverses.
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Third pair:
\( f(x) = 12x - 5 \)
\( g(x) = \frac{1}{12}x + 5 \)- \( f(g(x)) = f\left(\frac{1}{12}x + 5\right) = 12\left(\frac{1}{12}x + 5\right) - 5 = x + 60 - 5 = x + 55 \) (not equal to \( x \))
- \( g(f(x)) = g(12x - 5) = \frac{1}{12}(12x - 5) + 5 = x - \frac{5}{12} + 5 = x + \frac{55}{12} \) (not equal to \( x \))
Not inverses.
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Fourth pair:
\( f(x) = 11x - 9 \)
\( g(x) = 11x + 9 \)- \( f(g(x)) = f(11x + 9) = 11(11x + 9) - 9 = 121x + 99 - 9 = 121x + 90 \) (not equal to \( x \))
- \( g(f(x)) = g(11x - 9) = 11(11x - 9) + 9 = 121x - 99 + 9 = 121x - 90 \) (not equal to \( x \))
Not inverses.
After checking all pairs, it seems that none of the pairs provided are inverses of each other. Would you like to check the calculations or explore different functions?