5. construct a table of at least 4 ordered pairs of points on the graph of the following function and use…
5. construct a table of at least 4 ordered pairs of points on the graph of the following function and use the ordered pairs from the table to sketch the graph of the function.\n\n$$f(x) = \\begin{cases} 5 + x & \\text{if } x \\ge 1 \\\\ 2 & \\text{if } -2 \\le x < 1 \\\\ 1 - x^2 & \\text{if } x < -2 \\end{cases}$$
Answer
Explanation:
Step1: Select x-values for each piece
Choose $x = -3$ (bottom), $x = -1$ (middle), $x = 1$ (top), and $x = 2$ (top).
Step2: Calculate $f(x)$ for $x = -3$
Since $-3 < -2$, use $f(x) = 1 - x^{2}$. $$f(-3) = 1 - (-3)^{2} = 1 - 9 = -8$$
Step3: Calculate $f(x)$ for $x = -1$
Since $-2 \leq -1 < 1$, use $f(x) = 2$. $$f(-1) = 2$$
Step4: Calculate $f(x)$ for $x = 1$
Since $1 \geq 1$, use $f(x) = 5 + x$. $$f(1) = 5 + 1 = 6$$
Step5: Calculate $f(x)$ for $x = 2$
Since $2 \geq 1$, use $f(x) = 5 + x$. $$f(2) = 5 + 2 = 7$$
Answer:
The table of ordered pairs is:
| $x$ | $f(x)$ | Ordered Pair $(x, y)$ |
|---|---|---|
| $-3$ | $-8$ | $(-3, -8)$ |
| $-1$ | $2$ | $(-1, 2)$ |
| $1$ | $6$ | $(1, 6)$ |
| $2$ | $7$ | $(2, 7)$ |
To sketch the graph:
- For $x < -2$, plot the downward parabola $y = 1 - x^{2}$ (e.g., point $(-3, -8)$), ending with an open circle at $(-2, -3)$.
- For $-2 \leq x < 1$, plot a horizontal line segment at $y = 2$ from $x = -2$ (closed circle) to $x = 1$ (open circle).
- For $x \geq 1$, plot the line $y = 5 + x$ starting with a closed circle at $(1, 6)$ and passing through $(2, 7)$.