Question

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*Piece-Wise Defined Functions*

Answer 1

`\textsf{A.} \quad \begin{cases}-x -2 \quad \text{if}\;x < -2\\x + 4 \quad \;\;\;\text{if}\;x \geq -2\end{cases}`

Explanation:

When graphing piece-wise functions:

• An open circle indicates the value is not included in the interval.
• A closed circle indicates the value is included in the interval.
• An arrow show that the function continues indefinitely in that direction.

From inspection of the given graph:

• The equation for the line on the left side of the function has a negative slope and an x-intercept at x = -2.

• There is an open circle at x = -2 and the line continues to the left.
  ⇒ x < -2

• The equation for the line on the right side of the function has a positive slope and a y-intercept at y = 4.

• There is a closed circle at x = -2 and the line continues to the right.
  ⇒ x ≥ -2

Therefore, the definition of the function is:

`\begin{cases}-x + a \quad \text{if}\;x < -2\\x + 4 \quad \;\;\;\text{if}\;x \geq -2\end{cases}`

where a is some constant.

The only answer option that matches the given parameters is option A.

Answer 2

`{ \qquad\qquad\huge\underline{{\sf Answer}}}`

The function shown in the graph above is a discontinuous function, at x = -2

For values of x less than -2, the left side of the graph is applicable. whose equation is :

`\qquad \sf  \dashrightarrow \: y - 0 = \cfrac{1 - 0}{ - 3 - ( - 2)} (x - ( - 2))`

`\qquad \sf  \dashrightarrow \: y = \cfrac{1}{ - 3 + 2} (x + 2)`

`\qquad \sf  \dashrightarrow \: y = - 1(x + 2)`

`\qquad \sf  \dashrightarrow \: y = - x - 2 \:\: ; \:\:x<-2`

For values of x greater than -2, the right side of the graph is applicable. whose equation is :

`\qquad \sf  \dashrightarrow \: y - 4= \cfrac{4 - 2 }{0 - (-2)} (x -0)`

`\qquad \sf  \dashrightarrow \: y - 4 = \cfrac{2}{2} (x)`

`\qquad \sf  \dashrightarrow \: y - 4 = x`

`\qquad \sf  \dashrightarrow \: y = x + 4 \: \: ;\: \: x > - 2`

So, the correct choice will be Option " A "