Q44. - Algebra 2

Question

Write the function shown in the graph.

Step-by-Step Solution

Answer

The required function is $f\left(x\right)=\{\begin{array}{l}2ifx<-1\\ 2xif-1\le x\le 1\\ -xifx>1\end{array}$.

Step-1 &ndash; Piecewise function.

A function that is written using two or more expression is called a piecewise function.

Step-2 &ndash; Determine the function.

The given function is piecewise function as it consists different rays, segments and curves.

Step-3 &ndash; Determine the equation.

Consider the graph.

For $x<-1$  the function is the constant function $f\left(x\right)=2$.

For $-1\le x\le 1$  the function is a line with the slope of 2 and y-intercept 0, so $f\left(x\right)=2x$.

For $x>1$  the function is a line with the slope of $-1$  and passes through the point $(2,-2)$. 

Now use the point slope form to determine the equation of line.

 $\begin{array}{l}(y-y_1)=m(x-x_1)\\ (y-(-2\left)\right)=-1(x-2)\\ y=-x+2-2\\ y=-x\end{array}$.

 

Thus, $f\left(x\right)=-x$ for $x>1$

 

Hence, the function is $f\left(x\right)=\{\begin{array}{l}2ifx<-1\\ 2xif-1\le x\le 1\\ -xifx>1\end{array}$.