Determine a piecewise-defined function for the graph shown below.

A) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \ - ( x - 1 ) ^ { 2 } + 1 , & x 1 \end{array} \right.$$
B) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 0 \ - ( x - 1 ) ^ { 2 } + 1 , & x \leq 0 \end{array} \right.$$
C) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \geq 1 \ - x ^ { 2 } , & x \leq 1 \end{array} \right.$$
D) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \geq 0 \ - x ^ { 2 } , & x \leq 1 \end{array} \right.$$
E) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \ - ( x - 1 ) ^ { 2 } , & x 1 \end{array} \right.$$

Question

Determine a piecewise-defined function for the graph shown below.

A) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \ - ( x - 1 ) ^ { 2 } + 1 , & x > 1 \end{array} \right.$$ 
B) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 0 \ - ( x - 1 ) ^ { 2 } + 1 , & x \leq 0 \end{array} \right.$$ 
C) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \geq 1 \ - x ^ { 2 } , & x \leq 1 \end{array} \right.$$ 
D) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \geq 0 \ - x ^ { 2 } , & x \leq 1 \end{array} \right.$$ 
E) $$f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \ - ( x - 1 ) ^ { 2 } , & x > 1 \end{array} \right.$$

Quizplus · Accepted Answer

The graph consists of two segments. The left segment has a V-shape and passes through the point (1,1) with its vertex at (0,0). This suggests a function of the form f(x) = |x| for x ≤ 1. The right segment is a downward parabola that passes through the point (2,0) with its vertex at (1,1). This suggests a function of the form f(x) = −(x − 1)^2 + 1 for x > 1. Therefore, the piecewise-defined function that best fits the graph is:

$$f ( x ) = \left\{ \begin{array} { l l } 
| x | , & x \leq 1 \
- ( x - 1 ) ^ { 2 } + 1 , & x > 1
\end{array} \right.$$

Determine a Piecewise-Defined Function for the Graph Shown Below f(x)={∣x∣,x≤1−(x−1)2+1,x>1f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \\- ( x - 1 ) ^ { 2 } + 1 , & x > 1\end{array} \right.f(x)={∣x∣,−(x−1)2+1,​x≤1x>1​

Determine a Piecewise-Defined Function for the Graph Shown Below $f ( x ) = \left\{ \begin{array} { l l } | x | , & x \leq 1 \\- ( x - 1 ) ^ { 2 } + 1 , & x > 1\end{array} \right.$