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:

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

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

$f^{-1}(15)=f^{-1}(f(7))=7.$

And:

$(f(7))^{-1}=\frac{1}{f(7)}=\frac{1}{15}\text{.}$

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

$g^{-1}(2)=g^{-1}(g(-6))=-6.$

And:

$(g(-6))^{-1}=\frac{1}{g(-6)}=\frac{1}{2}\text{.}$

Answer:

A)

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

B)

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