If `p(x)=|2-x|+(1)/(2)`, which of the following could be the graph of `y = p (x)` ?

Options

A.
B.
C.
D.

Answer

A

Explanation

The question says `p(x)=|2-x|+(1)/(2)` and asks for the graph of `y = p(x)` - meaning the graph of `y=|2-x|+(1)/(2)`.
The smallest that y can be is `(1)/(2)`, because the absolute value of anything is nonnegative. That narrows the choices to (A) and (C ). Notice that (D) cannot be functions, because for some values of x they show two associated values of y. A function, by definition, yields no more than one y for any particular x.
To choose between (A) and (C ), think about what value of x yields `y = (1)/(2)` .
`|2-x|+(1)/(2)=(1)/(2)`
`|2-x|=0`
`2-x=0`
`2=x`
So the graph reaches its minimum point at x = 2, which means the answer must be (A).