Question

Which of the following transformations in order will give $-|(-x)+3|$ from $|x|$?
(1) $f\left(x\right)\to f(-x)$
(2) $f\left(x\right)\to f(x+3)$
(3) $f\left(x\right)\to -f\left(x\right)$

(4) $f\left(x\right)\to f(x-3)$

• A.
• B.
• C.
• D.

Answer 1

Answer: (B), (C), (D)

The correct options are
B

C

D

We will go through the options and check if we get $-|(-x)+3|$ from $|x|$.
A. (1)$\to \left(2\right)\to \left(3\right)\to \left(4\right)$

(1) $\Rightarrow |x|\to |-x|$
(2) $\Rightarrow |-x|\to |-(x+3)|$
(3) $\Rightarrow |-(x+3)|\to -|-(x+3)|=-|-x-3|$
(4) $\Rightarrow -|-x-3|\to -|-(x-3)-3|$
= $-|-x+3-3|$
= $-|-x|$
We got ­$-|-x|$. this is not want we want.

B. (2)$\to \left(3\right)\to \left(1\right)$
(2) $\Rightarrow |x|\to |x+3|$
(3) $\Rightarrow |x+3|\to -|x+3|$
(1) $\Rightarrow -|x+3|\to -|-x+3|$
This gives us the required function.

C. (3)$\to \left(2\right)\to \left(1\right)$
(3) $\Rightarrow |x|\to -|x|$
(2) $\Rightarrow -|x|\to -|x+3|$
(1) $\Rightarrow -|x+3|\to -|-x+3|$
(C) also gives us the required function.
(D) (2)$\to \left(1\right)\to \left(3\right)$
(2) $\Rightarrow |x|\to |x+3|$

(1) $\Rightarrow |x+3|\to |-x+3|$
(3) $\Rightarrow |-x+3|\to -|-x+3|$
(D) also gives us the required function. This is why we choose to go through the options rather than deducting from the question. More than one type of transformation can give the same function.