Comparing Characteristics of Functions — Quiz

Question 37 • Mathematics II

Which function has a domain and range that includes all real values?

Answer

Image with description On a coordinate plane, the x and y axes range from negative 8 to 8 in increments of 2. The graph has an inverted v-shaped curve with a vertex at the origin. One line extends infinitely into the third quadrant passing through (negative 4 comma negative 4). The other line extends infinitely into the fourth quadrant passing through (4 comma negative 4).

Image with description On a coordinate plane, the x-axis ranges from negative 6 to 10 in increments of 2, and the y-axis ranges from negative 10 to 8 in increments of 2. The graph has a straight line with a positive slope, goes through (0 comma negative 4), and (2 comma 0). It extends infinitely in both directions.

Image with description On a coordinate plane, the x and y axes range from negative 8 to 8 in increments of 2. The graph has a parabola that opens up with a vertex at the origin. It passes through (negative 2 comma 3) and (2 comma 3).

Image with description On a coordinate plane, the x-axis ranges from negative 6 to 10 in increments of 2, and the y-axis ranges from negative 6 to 10 by 2. A smooth curve starts from (2 comma 2), goes up, and passes through (6 comma 4).

Explanation

A function has a domain and range that include all real values if its graph extends infinitely in both directions and takes every possible y-value. Choice D is a nonhorizontal line that extends infinitely in both directions, so its x-values and y-values are both unrestricted. The other graphs have a restricted domain or range, such as the parabola in B, whose outputs are only zero or greater.

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