Question

which key features do these representations of linear functions have in common? select all that apply.

(chart and image omitted as per instruction)

both functions are undefined or continuous
increasing over the domain
decreasing over the domain
x - intercept of (2, 0)

Explanation:

Step1: Analyze continuity of functions

Linear functions (both graphed and tabulated, as the table represents a linear relationship) are continuous over their entire domain, so both are continuous.

Step2: Check if functions are increasing

For the graph:

The line rises from left to right, so it has a positive slope. Calculate slope using x-intercept $(-4,0)$ and y-intercept $(0,2)$:
$\text{Slope} = \frac{2-0}{0-(-4)} = \frac{2}{4} = 0.5$

For the table:

Use two points $(2,2.5)$ and $(4,3)$ to find slope:
$\text{Slope} = \frac{3-2.5}{4-2} = \frac{0.5}{2} = 0.25$
Both slopes are positive, so functions are increasing.

Step3: Verify x-intercept

The graph's x-intercept is $(-4,0)$, not $(2,0)$. For the table's function, set $y=0$: first find its equation $y=0.25x+2$. Solve $0=0.25x+2$:
$0.25x = -2 \implies x = -8$, so its x-intercept is $(-8,0)$.

Step4: Rule out decreasing option

Both slopes are positive, so functions are not decreasing.

Answer:

• Both functions are undoubtedly continuous
• Increasing over the domain