To determine which two functions are inverses of each other, we can use function composition. If \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( f \) and \( g \) are inverses.
Let's check the pairs of functions provided.
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First pair:
- \( f(x) = 13x + 9 \)
- \( g(x) = 3x - 9 \)
Calculate \( f(g(x)) \): \[ f(g(x)) = f(3x - 9) = 13(3x - 9) + 9 = 39x - 117 + 9 = 39x - 108 \]
Calculate \( g(f(x)) \): \[ g(f(x)) = g(13x + 9) = 3(13x + 9) - 9 = 39x + 27 - 9 = 39x + 18 \]
Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), these are not inverses.
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Second pair:
- \( f(x) = 2x + 7 \)
- \( g(x) = 7x + 2 \)
Calculate \( f(g(x)) \): \[ f(g(x)) = f(7x + 2) = 2(7x + 2) + 7 = 14x + 4 + 7 = 14x + 11 \]
Calculate \( g(f(x)) \): \[ g(f(x)) = g(2x + 7) = 7(2x + 7) + 2 = 14x + 49 + 2 = 14x + 51 \]
Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), these are not inverses.
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Third pair:
- \( f(x) = 3x + 5 \)
- \( g(x) = \frac{x - 5}{3} \)
Calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x - 5}{3}\right) = 3\left(\frac{x - 5}{3}\right) + 5 = x - 5 + 5 = x \]
Calculate \( g(f(x)) \): \[ g(f(x)) = g(3x + 5) = \frac{(3x + 5) - 5}{3} = \frac{3x}{3} = x \]
Since \( f(g(x)) = x \) and \( g(f(x)) = x \), these are inverses.
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Fourth pair:
- \( f(x) = 2x + 7 \)
- \( g(x) = \frac{x + 7}{2} \)
Calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x + 7}{2}\right) = 2\left(\frac{x + 7}{2}\right) + 7 = x + 7 + 7 = x + 14 \]
Calculate \( g(f(x)) \): \[ g(f(x)) = g(2x + 7) = \frac{(2x + 7) + 7}{2} = \frac{2x + 14}{2} = x + 7 \]
Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), these are not inverses.
Conclusion: The only pair that are inverses of each other is:
- \( f(x) = 3x + 5 \) and \( g(x) = \frac{x - 5}{3} \)