Question

graph the following piecewise function, determine its range and evaluate $f(4)$.
$f(x)=\

$$\begin{cases}-4x - 3&x < 0\\\\2x - 1&x \\geq 0\\end{cases}$$

$

Explanation:

Step1: Identify relevant piece for $f(4)$

Since $4 \geq 0$, use $f(x)=2x-1$.

Step2: Substitute $x=4$ into the formula

$f(4)=2(4)-1$

Step3: Calculate the result

$f(4)=8-1=7$

Step4: Graph the first piece ($x<0$)

For $f(x)=-4x-3$, plot points e.g., $x=-1$: $f(-1)=-4(-1)-3=1$, $x=0$: $f(0)=-3$ (open circle at $(0,-3)$), draw the line for $x<0$.

Step5: Graph the second piece ($x\geq0$)

For $f(x)=2x-1$, plot points e.g., $x=0$: $f(0)=-1$ (closed circle at $(0,-1)$), $x=4$: $f(4)=7$, draw the line for $x\geq0$.

Answer:

$f(4)=7$
(Graph: A line $y=-4x-3$ for all $x<0$ with an open circle at $(0,-3)$, and a line $y=2x-1$ for all $x\geq0$ with a closed circle at $(0,-1)$, passing through $(4,7)$)