Question

How to find an inverseF(x) = 3x - 1 solve for Y

Answer 1

Answer:

f^-1(x) = x/3 + 1/3

Explanation:

We were given that:

$f(x)=3x-1$

We are to calculate the inverse of the function above. We will do so by following the steps enumerated below:

I. We will replace f(x) with ''y'', we have:

$\begin{gathered} f(x)=3x-1 \\ f(x)=y \\ \Rightarrow y=3x-1 \\ \\ \therefore y=3x-1 \end{gathered}$

II. We will replace the position of ''x'' with ''y'' and ''y'' with ''x''. We have:

$\begin{gathered} y=3x-1\rightarrow x=3y-1 \\ x=3y-1 \\ \\ \therefore x=3y-1 \end{gathered}$

III. We will proceed to make ''y'' the subject of the formula, we have:

$\begin{gathered} x=3y-1 \\ \text{Add ''1'' to both sides, we have:} \\ x+1=3y\Rightarrow3y=x+1 \\ 3y=x+1 \\ \text{Divide through by ''3'' to obtain ''y'', we have:} \\ y=\frac{1}{3}x+\frac{1}{3} \\ \\ \therefore y=\frac{1}{3}x+\frac{1}{3} \end{gathered}$

IV. We will now replace ''y'' with the inverse symbol, we have:

$\begin{gathered} y=\frac{1}{3}x+\frac{1}{3} \\ y\rightarrow f^{-1}(x) \\ f^{-1}(x)=\frac{1}{3}x+\frac{1}{3} \\ \\ \therefore f^{-1}(x)=\frac{1}{3}x+\frac{1}{3} \end{gathered}$

We will now proceed to verify the answer obtained in IV. above. We have:

$\begin{gathered} \mleft({f\circ{f^{-1}}}\mright)\mleft(x\mright)=f\lbrack f^{-1}(x)\rbrack \\ ({f\circ{f^{-1}}})(x)=f(\frac{1}{3}x+\frac{1}{3}) \\ ({f\circ{f^{-1}}})(x)=3(\frac{1}{3}x+\frac{1}{3})-1 \\ ({f\circ{f^{-1}}})(x)=x+1-1 \\ ({f\circ{f^{-1}}})(x)=x \\ \\ \therefore The\text{ answer obtained in IV. is correct} \end{gathered}$

Therefore, the inverse of the function is: f^-1(x) = x/3 + 1/3