Marin writes the functions m times x is equal to the fraction with numerator 3 x and denominator x plus 7 and n times x is equal to the fraction with numerator 7 x and denominator 3 minus x Which equation must be true for m times x and n times x to be inverse functions?
Answer
A
the fraction with numerator 3 times open paren the fraction with numerator 7 x and denominator 3 minus x close paren and denominator open paren the fraction with numerator 7 x and denominator 3 minus x close paren plus 7 is equal to the fraction with numerator 7 times open paren the fraction with numerator 3 x and denominator x plus 7 close paren and denominator 3 minus open paren the fraction with numerator 3 x and denominator x plus 7 close paren is equal to x
B
the fraction with numerator 3 times open paren the fraction with numerator 3 x and denominator x plus 7 close paren and denominator open paren the fraction with numerator 3 x and denominator x plus 7 close paren plus 7 is equal to the fraction with numerator 7 times open paren the fraction with numerator 7 x and denominator 3 minus x close paren and denominator 3 minus open paren the fraction with numerator 7 x and denominator 3 minus x close paren is equal to 0
C
the fraction with numerator 3 times open paren the fraction with numerator 7 x and denominator 3 minus x close paren and denominator open paren the fraction with numerator 7 x and denominator 3 minus x close paren plus 7 is equal to the fraction with numerator 7 times open paren the fraction with numerator 3 x and denominator x plus 7 close paren and denominator 3 minus open paren the fraction with numerator 3 x and denominator x plus 7 close paren is equal to 0
D
the fraction with numerator 3 times open paren the fraction with numerator 3 x and denominator x plus 7 close paren and denominator open paren the fraction with numerator 3 x and denominator x plus 7 close paren plus 7 is equal to the fraction with numerator 7 times open paren the fraction with numerator 7 x and denominator 3 minus x close paren and denominator 3 minus open paren the fraction with numerator 7 x and denominator 3 minus x close paren is equal to x
Explanation
For inverse functions, both compositions must equal x: m(n(x)) = x and n(m(x)) = x. Substituting the given functions gives 3(7x/(3 − x)) / (7x/(3 − x) + 7) = x and 7(3x/(x + 7)) / (3 − 3x/(x + 7)) = x, which is the equation in choice A.
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