Question

For each pair of functions and below, find and .

Answer 1

Step-by-step explanation:

Given the functions:

`\begin{gathered} f(x)=-(1)/(4x) \\ g(x)=(1)/(4x) \end{gathered}`

In order to know if the functions are inverses of each other, we must show that f(g(x)) is equal to g(f(x)).

Get the composite function f(g(x))

`\begin{gathered} f(g(x))=f((1)/(4x)) \\ \end{gathered}`

To get f(g(x)), we will replace the variable "x" in f(x) with 1/4x as shown:

`\begin{gathered} f((1)/(4x))=-(1)/(4((1)/(4x))) \\ f((1)/(4x))=-(1)/((4)/(4x)) \\ f((1)/(4x))=-(1)/((1)/(x)) \\ f((1)/(4x))=-x \\ f(g(x))=-x \end{gathered}`

Next is to get the composite function g(f(x))

`\begin{gathered} g(f(x))=g(-(1)/(4x)) \\ \end{gathered}`

To get g(f(x)), we will replace the variable "x" in g(x) with -1/4x as shown:

`\begin{gathered} g(-(1)/(4x))=(1)/(4(-(1)/(4x))) \\ g(-(1)/(4x))=(1)/(-(4)/(4x)) \\ g(-(1)/(4x))=(1)/(-(1)/(x)) \\ g(-(1)/(4x))=-x \\ g(f(x))=-x \end{gathered}`

From the solution above, since f(g(x)) = g(f(x)) = -x, hence the functions f(x) and g(x) are