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Determine which two functions are inverses of each other by function composition.(1 point)Responses

f(x)=15x+2 and g(x)=2x−15 f(x)=−4x−1 / 3 and g(x)=−3x−1/ 4 f(x)=1/11x+2 and g(x)=11x−2 f(x)=3x+7/5 and g(x)=3/5x−7

1 answer

To determine which pairs of functions are inverses of each other, we will perform function composition on each pair (i.e., \( f(g(x)) \) and \( g(f(x)) \)). If both compositions result in \( x \), then the functions are inverses of each other.

Pair 1:

\( f(x) = 15x + 2 \) and \( g(x) = 2x - 15 \)

  1. Calculate \( f(g(x)) \): \[ f(g(x)) = f(2x - 15) = 15(2x - 15) + 2 = 30x - 225 + 2 = 30x - 223 \] (Not equal to \( x \))

  2. Calculate \( g(f(x)) \): \[ g(f(x)) = g(15x + 2) = 2(15x + 2) - 15 = 30x + 4 - 15 = 30x - 11 \] (Not equal to \( x \))

Pair 2:

\( f(x) = -4x - \frac{1}{3} \) and \( g(x) = -3x - \frac{1}{4} \)

  1. Calculate \( f(g(x)) \): \[ f(g(x)) = f(-3x - \frac{1}{4}) = -4(-3x - \frac{1}{4}) - \frac{1}{3} = 12x + 1 - \frac{1}{3} \] \[ = 12x + \frac{3 - 1}{3} = 12x + \frac{2}{3} \] (Not equal to \( x \))

  2. Calculate \( g(f(x)) \): \[ g(f(x)) = g(-4x - \frac{1}{3}) = -3(-4x - \frac{1}{3}) - \frac{1}{4} = 12x + 1 - \frac{1}{4} \] \[ = 12x + \frac{4 - 1}{4} = 12x + \frac{3}{4} \] (Not equal to \( x \))

Pair 3:

\( f(x) = \frac{1}{11}x + 2 \) and \( g(x) = 11x - 2 \)

  1. Calculate \( f(g(x)) \): \[ f(g(x)) = f(11x - 2) = \frac{1}{11}(11x - 2) + 2 = x - \frac{2}{11} + 2 \] \[ = x - \frac{2}{11} + \frac{22}{11} = x + \frac{20}{11} \] (Not equal to \( x \))

  2. Calculate \( g(f(x)) \): \[ g(f(x)) = g(\frac{1}{11}x + 2) = 11(\frac{1}{11}x + 2) - 2 = x + 22 - 2 = x + 20 \] (Not equal to \( x \))

Pair 4:

\( f(x) = \frac{3}{5}x + \frac{7}{5} \) and \( g(x) = \frac{3}{5}x - 7 \)

  1. Calculate \( f(g(x)) \): \[ f(g(x)) = f(\frac{3}{5}x - 7) = \frac{3}{5}(\frac{3}{5}x - 7) + \frac{7}{5} = \frac{9}{25}x - \frac{21}{5} + \frac{7}{5} \] \[ = \frac{9}{25}x - \frac{21}{5} + \frac{7}{5} = \frac{9}{25}x - \frac{14}{5} \] (Not equal to \( x \))

  2. Calculate \( g(f(x)) \): \[ g(f(x)) = g(\frac{3}{5}x + \frac{7}{5}) = \frac{3}{5}(\frac{3}{5}x + \frac{7}{5}) - 7 = \frac{9}{25}x + \frac{21}{5} - 7 \] \[ = \frac{9}{25}x + \frac{21}{5} - \frac{35}{5} = \frac{9}{25}x - \frac{14}{5} \] (Not equal to \( x \))

Conclusion

None of the pairs satisfy the condition of being inverses.