Question

If the function $f\left(x\right)=ax+b$ has its own inverse then the ordered pair (a,b) can be

• A. (1,0)
• B. (-1,0)
• C. (-1,1)
• D. (1,1)

Answer 1

Answer: (A), (B), (C)

The correct options are
A (1,0)
B (-1,0)
C (-1,1)

$f\left(x\right)=a\left(x\right)+b$
We know that $f\left(f^{-1}\right(x\left)\right)=x$ and it is given that $f\left(x\right)$ is its own inverse.
Hence
$f\left(f^{-1}\right(x\left)\right)=a(ax+b)+b$
$=a^{2}x+b(a+1)$
$=x$
By comparing coefficients, we get
$a^2=1$
$a=\pm 1$
And $b(1+a)=0$
$b=0$ or $a=-1$
Hence the ordered pairs of (a,b) are
$(-1,1)$, $(-1,0)$, $(1,0)$