A function is defined as shown.f(x)=−3x+4What is the inverse of f(x)?A. f^−1(x)=−1/3x−4/3B. f − 1 ( x ) = − 1/3 x − 4 C. f^−1(x)=−1/3x+4/3D. f ^− 1 ( x ) = − 1/3x +4

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04-11-2022
Mathematics

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A function is defined as shown.f(x)=−3x+4What is the inverse of f(x)?A. f^−1(x)=−1/3x−4/3B. f − 1 ( x ) = − 1/3 x − 4 C. f^−1(x)=−1/3x+4/3D. f ^− 1 ( x ) = − 1/3x +4

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SenecaB492907 SenecaB492907 · Answer 1 · 2022-11-04T02:09:50-04:00

We need to find the inverse function of:

[tex]f\mleft(x\mright)=−3x+4[/tex]

In order to do so, we can:

• replace x with y;

,

• replace f(x) with x;

,

• isolate y on the left side;

,

• replace y with f⁻¹(x).

We obtain:

[tex]\begin{gathered} x=-3y+4 \\ \\ x+3y=-3y+4+3y \\ \\ 3y+x=4 \\ \\ 3y+x-x=4-x \\ \\ 3y=4-x \\ \\ \frac{3}{3}y=\frac{4-x}{3} \\ \\ y=-\frac{1}{3}x+\frac{4}{3} \\ \\ f^{-1}\left(x\right)=-\frac{1}{3}x+\frac{4}{3} \end{gathered}[/tex]

Answer: option C

[tex]f^{-1}(x)=-\frac{1}{3}x+\frac{4}{3}[/tex]

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