Question

Which is the correct piecewise definition for the function |x−3|=y−1?

A. y=−x+2 for x≥−3 and y=x−4 for x<−3

B. y=x−2 for x≥3 and y=−x+4 for x<3

C. y=x−2 for x<3 and y=−x+4 for x≥3

D. y=−x+2 for x<−3 and y=x−4 for x≥−3

Answer 1

B.
`y= x-2\; for\; x\geq3\; and\; y=-x+4 \;for \;x<3`

Step-by-step explanation:

Given function

`\mid{ x-3}\mid=y-1`

We know that the break of modulus function

f(x)=
`\mid{x-1} \mid`

f(x)

= x-1 for
`x\geq 1`

And
`f(x)=-(x-1)\; for\;x<1`

Therefore, similarly we break the modulus function in the same way

`y-1=\mid{x-3}\mid`

we can write as

`y-1= x-3 for\; x\geq3`

Therefore ,
`y=x-2 \;for\;x\geq 3` ( by using subtraction property of equality )

And
`y-1=-x+3\;for\;x<3`

We can write as
`y=-x+4 \;for\; x<3` (By simplication)

Hence, B.
`y=x-2\; for\; x\geq 3\; and \; y=-x+4\; for\; x<3` is correct option .

Answer 2

B. y=x−2 for x≥3 and y=−x+4 for x<3

Step-by-step explanation:

Answer choices B and C are the only ones with the breakpoint at x=3, where the absolute value function has an argument of zero.

Of those, answer choice B is the only one with a positive slope for x > 3, so is the only correct choice.