Question

Finding Piecewise Defined Functions A graph of a piecewise defined function is given. Find a formula for the function in the indicated form.

Short answer

$f\left(x\right)=\left\{\begin{array}{l}1\text{if}x\le -1\\ 1-x\text{if}-1<x\le 2\\ -2\text{if}x>2\end{array}\right.$

Step-by-step solution

Step 1. Given information.

Given that, the graph is-

Step 2. To find the value of function when x≤−1.

From the graph, we can see that,

When $x\le -1$, then the values taken by x are $x=-1,-2,-3,...$

And corresponding to these x-values, the y-values are 1, 1, 1, …

Therefore, we get

$f\left(x\right)=1\text{if}x\le -1$

Step 3. To find the value of function when −1<x≤2.

From the graph, we can see that,

When $-1<x\le 2$, then the values taken by x are $x=0,1,2$

And corresponding to these x-values, the y-values are 1, 0, -1

Therefore, we get

The function $f\left(x\right)$is passes through the points$\left(0,1\right),\left(1,0\right),\left(2,-1\right)$

Now,

By taking points $\left(0,1\right),\left(1,0\right)$and using Slope-Intercept form, we get

role="math" localid="1644920287023" $\begin{array}{l}y-1=\frac{0-1}{1-0}\left(x-0\right)\\ \Rightarrow y-1=-1\left(x\right)\\ \Rightarrow y-1=-x\\ \Rightarrow y=1-x\end{array}$

Therefore, we get

$f\left(x\right)=1-x\text{if-1<}x\le 2$

Step 4. To find the value of function when x>2.

From the graph, we can see that,

When $x>2$, then the values taken by x are $x=3,4,5,...$

And corresponding to these x-values, the y-values are -2, -2, -2, …

Therefore, we get

$f\left(x\right)=-2\text{if}x>2$

Step 5. To find the formula for the function.

By using Step 2, 3 and 4, we can conclude that the formula for the function be

$f\left(x\right)=\left\{\begin{array}{l}1\text{if}x\le -1\\ 1-x\text{if}-1<x\le 2\\ -2\text{if}x>2\end{array}\right.$