Question

what is the inverse of
f(x)= 8/27x^3

Answer 1

\(\bf f(x)=\cfrac{8}{27x^3}\quad ?\)

Answer 2

no its more like \[f(x)=\frac{ 8 }{27}x ^{3}\]

Answer 3

ok... well, to get the inverse "relation" of the function, what we simply do is "swap about" the variables, and then solve for "y"

that is \(\bf f(x)={\color{brown}{ y}}=\cfrac{8}{27}{\color{olive}{ x}}^3\qquad inverse\implies {\color{olive}{ x}}=\cfrac{8}{27}{\color{brown}{ y}}^3\)

then solve for "y"

Answer 4

just swap x and y then make y the subject

so

\[x = \frac{8}{27} y^3\]

now make y the subject

Answer 5

thats how far i got, i wasnt sure how to start off solving it @campbell_st @jdoe0001

Answer 6

ok... so multiply both sides by 27

divide both sides by 8

then take the cube root of both sides

and

8 and 27 are both cubic numbers

Answer 7

oh okay thank you i was trying to take the cubed root first @campbell_st

Answer 8

\(\bf f(x)={\color{brown}{ y}}=\cfrac{8}{27}{\color{olive}{ x}}^3\qquad inverse\implies {\color{olive}{ x}}=\cfrac{8}{27}{\color{brown}{ y}}^3
\\ \quad \\
x=\cfrac{8y^3}{27}\implies 27x=8y^3\implies \cfrac{27x}{8}=y^3
\\ \quad \\
\implies \Large \sqrt[3]{\cfrac{27x}{8}}=y\iff f^{-1}(x)\)

Answer 9

or

\[y = \frac{3}{2} \sqrt[3]{x}\]

Answer 10

\(\bf \sqrt[3]{\cfrac{27x}{8}}\implies \cfrac{\sqrt[3]{27x}}{\sqrt[3]{8}}\implies \cfrac{{\color{red}{ \sqrt[3]{27}}}\cdot \sqrt[3]{x}}{{\color{red}{ \sqrt[3]{8}}}}\)

notice as shown above by campbell_st

Answer 11

okay thank ya'll @jdoe0001 @campbell_st

Answer 12

yw