Question

O GRAPHS AND FUNCTIONSDetermining whether two functions are inverses of each other

Answer 1

Technically, for f ( x ) f\left( x \right) f(x) and g ( x ) g\left( x \right) g(x) to be inverses of each other, you must show that function composition works both ways! Therefore, the composition of function f with g equals

Answer 2

1) Let's begin with that by finding the composite functions.

2) So, let's begin with the ones to the left:

`\begin{gathered} (f(g(x))=(1)/(2\cdot(1)/(2x))=(1)/((1)/(x))=x \\ \\ (g(f(x))=(1)/(2\cdot(1)/(2x))=(1)/((1)/(x))=x \\ ----- \\ \\ f^(x)=(1)/(2x) \\ \\ y=(1)/(2x) \\ \\ x=(1)/(2y) \\ \\ 2yx=1 \\ \\ 2y=(1)/(x) \\ \\ y=(1)/(x)/\text{ 2} \\ \\ y=(1)/(2x) \end{gathered}`

Hence, we can tell that:

3) Let's now proceed with that one on the right:

`\begin{gathered} f\left(g\left(x\right)\right)=x+3+3\Rightarrow f(g(x))=x+6 \\ \\ g(f(x))=(x+3)+3\Rightarrow g(f(x))=x+6 \end{gathered}`

And now the inverse:

`\begin{gathered} f(x)=x+3 \\ \\ y=x+3 \\ \\ x=y+3 \\ \\ x-y=3 \\ \\ -y=3-x \\ \\ y=x-3 \\ \\ f^(-1)(x)=x-3 \end{gathered}`

And then, the answer is: