Question

practice: piecewise functions
1
evaluate:
$f(-4) =$
$f(-3) =$
$f(-1) =$
$f(2) =$
zeros:
$y$-intercept:
domain:
range:
extrema:
inc/dec/constant:
continuity:
end behavior:
equation:
$f(x) = \

$$\begin{cases} \\underline{\\quad} |x \\underline{\\quad}| \\underline{\\quad}, \\underline{\\quad\\quad\\quad\\quad} \\\\ \\underline{\\quad} (x \\underline{\\quad})^2 \\underline{\\quad}, \\underline{\\quad\\quad\\quad\\quad} \\end{cases}$$

$

Explanation:

Step1: Find f(-4)

Check x=-4 on the graph: closed dot at y=3.

Step2: Find f(-3)

Check x=-3 on the graph: y=4.

Step3: Find f(-1)

Check x=-1 on the graph: y=1.

Step4: Find f(2)

Check x=2 on the graph: y=-3.

Step5: Identify zeros

Zeros are x where y=0: x=1, x≈-0.5.

Step6: Find y-intercept

y-intercept at x=0: y=0.5.

Step7: Determine domain

All x from -4 to ∞: [-4, ∞).

Step8: Determine range

All y ≥ -3: [-3, ∞).

Step9: Find extrema

Minimum at (2, -3).

Step10: Check inc/dec/constant

Dec: [-4,2); Inc: (2,∞).

Step11: Check continuity

Continuous on [-4, ∞).

Step12: End behavior

As x→∞, f(x)→∞; as x→-4+, f(x)=3.

Step13: Write piecewise equation

Left piece: line from (-4,3) to (2,-3): slope = (-3-3)/(2-(-4))=-1, equation y=-x-1 for x≤2. Right piece: parabola vertex (2,-3): y=(x-2)²-3 for x>2.

Answer:

f(-4)=3
f(-3)=4
f(-1)=1
f(2)=-3
Zeros: x≈-0.5, x=1
y-intercept: 0.5
Domain: [-4, ∞)
Range: [-3, ∞)
Extrema: Minimum at (2, -3)
Inc/Dec/Constant: Dec on [-4,2), Inc on (2,∞)
Continuity: Continuous on [-4, ∞)
End Behavior: As x→∞, f(x)→∞; as x→-4+, f(x)=3
Equation: \( f(x)=

$$\begin{cases}-x-1, & x\leq2 \\ (x-2)^2-3, & x>2\end{cases}$$

\)