Question

If $f\left(x\right)=3x-1$, n represents the slope of the line with the equation $y=f^{-1}\left(x\right)$, and p represents the slope of a line that is perpendicular to the line with the equation $y=f\left(x\right)$ then find the value of $np$.

• A. $-9$
• B. $-\frac{1}{9}$
• C. $\frac{1}{9}$
• D. $9$
• E. It cannot be determined from the information given

Answer 1

The correct option is B $-\frac{1}{9}$

Given, $f\left(x\right)=3x-1$

Then $y=3x-1$

So slope $3$

Then slope of line perpendicular line $y=3x-1$ is $p=-\frac{1}{3}$

Change $(x,y)$ from $(y,x)$ in equation $f\left(x\right)=3x-1$

Then $x=3y-1$

$\Rightarrow 3y=x+1$

$\Rightarrow y=\frac{1}{3}-\frac{1}{3}$

$\Rightarrow f^{-1}\left(x\right)=\frac{1}{3}x+\frac{1}{3}$

The slope of $f^{-1}\left(x\right)$ is $n=\frac{1}{3}$

Then $np=\frac{-1}{3}\times \frac{1}{3}=-\frac{1}{9}$