Question

find the inverse of (f)x= 7x - 3. what is the value of the composition of the function and its inverse

Answer 1

hmmm did you find the inverse?

Answer 2

no idk how

Answer 3

\(\bf f(x)= {\color{red}{ y}}=7{\color{blue}{ x}} - 3\qquad inverse\implies {\color{blue}{ x}}=7{\color{red}{ y}} - 3\)
to find the inverse relation of the function, as you can see, we simply switch about the variables
then solve for "y", what does that give you?

Answer 4

would it be (x+3)/7= y ?

Answer 5

yeap

Answer 6

thank you!

Answer 7

\(\bf f(x)= {\color{red}{ y}}=7{\color{blue}{ x}} - 3\qquad inverse\implies {\color{blue}{ x}}=7{\color{red}{ y}} - 3\\\quad\\\implies \cfrac{x+3}{7}=y=f^{-1}(x)\)

so now "what is the value of the composition of the function and its inverse" ?

Answer 8

I have no idea

Answer 9

one sec

Answer 10

so the composition of f(x) and it's inverse function will be \(\bf f(x)= {\color{red}{ y}}=7{\color{blue}{ x}} - 3\qquad inverse\implies {\color{blue}{ x}}=7{\color{red}{ y}} - 3
\\ \quad \\
\implies {\color{green}{ \cfrac{x+3}{7}}}=y=f^{-1}(x)
\\ \quad \\

f(\quad f^{-1}(x)\quad )=7x - 3\implies f\left({\color{green}{ \cfrac{x+3}{7}}}\right)=
\cancel{7}\left({\color{green}{ \cfrac{x+3}{\cancel{7}}}}\right)-3
\\ \quad \\
f\left({\color{green}{ \cfrac{x+3}{7}}}\right)=x\cancel{+3-3}\)

Answer 11

keep in mind that \(\bf \textit{composition of function and inverse is always }=x
\\ \quad \\
\textit{that is}\qquad f(f^{-1}({\color{blue}{ x}}))={\color{blue}{ x}}\)

Answer 12

is that the answer?

Answer 13

yes

Answer 14

thank you so much!

Answer 15

yw