Question

A) If f is one-to-one and f(7) = 15, then f^-1(15) = ______ and (f(7))^-1 = _______. B) If g is one-to-one and g(-6) = 2, then g^-1(2) = _______ and (g(-6))^-1 = ________.

Answer 1

Recall that, if h(x) is a one-to-one function, then:

`\begin{gathered} h^(-1)(h(x))=x\text{.} \\ (h(x))^(-1)=(1)/(h(x))\text{.} \end{gathered}`

A) Since f is a one-to-one function and f(7)=15, then:

`f^(-1)(15)=f^(-1)(f(7))=7.`

And:

`(f(7))^(-1)=(1)/(f(7))=(1)/(15)\text{.}`

B) Since g is a one-to-one function and g(-6)=2, then:

`g^(-1)(2)=g^(-1)(g(-6))=-6.`

And:

`(g(-6))^(-1)=(1)/(g(-6))=(1)/(2)\text{.}`

Answer:

A)

`\begin{gathered} f^(-1)(15)=7, \\ (f(7))^(-1)=(1)/(15)\text{.} \end{gathered}`

B)

`\begin{gathered} g^(-1)(2)=-6, \\ (g(-6))^(-1)=(1)/(2)\text{.} \end{gathered}`