Question

Consider the following graph of a piecewise-defined function.
Choose the correct expression to represent the piecewise defined function.
1.x<-2,f(x)=
A.-3
B.5
C.2
2.-2 A.x+2
B.2x+2
C.2x
3.x>3,f(x)=
A.-3
B.5
C.2
Help please.

Answer 1

Answer:

$f(x)=\left\{ \begin{array}{c}5\:If\:-\infty<x<-2\\2x+2\:If\:-2\leq x<3\\-3\:If\:x\geq3\end{array}\right.$

Step-by-step explanation:

A piecewise-defined function is one that you define not by a single equation, but by two or more. From the figure, we know that this piecewise-defined function is defined by three equations, so our goal is to find each. Two of these equations are constant functions while the other equation is a linear function. So:

1. FOR THE FIRST CONSTANT FUNCTION:

$f(x)=5, \ if \ -\infty<x<-2$

Keep in mind that for this constant function at x = 2, it isn't defined, that's why we choose the symbol <

2. FOR THE LINEAR FUNCTION:

$y=mx+b \\ \\ b=2 \\ \\ m=\frac{2-0}{0-(-1)}=2 \\ \\ y=2x+2 \\ \\ f(x)=2x+2 \ if \ -2 \leq x<3$

Keep in mind that for this linear function at x = 3, it isn't defined, that's why we use the symbol <

3. FOR THE SECOND CONSTANT FUNCTION:

$f(x)=-3, \ if \ x\geq 3$

Keep in mind that for this constant function at x = 3, it is defined, that's why we choose the symbol ≥

In conclusion, the function is:

$f(x)=\left\{ \begin{array}{c}5\:If\:-\infty<x<-2\\2x+2\:If\:-2\leq x<3\\-3\:If\:x\geq3\end{array}\right.$