# How do you determine the domain and range of a graph inequalities?

Apr 1, 2015

To determine the domain is the same as to determine which numbers appear as the first number (the x-value) in an ordered pair that is part of the graph.
To determine the range is the same as to determine which numbers appear as the second number (the y-value) in an ordered pair that is part of the graph.

Here are some examples:

$y \ge {x}^{2} + 3$

graph{y >= x^2+3 [-11.6, 13.72, 0.15, 12.81]}

Although it is not 100% certain from just the graph, this graph does get wider and wider. Every $x$ does appear in some ordered pair on the graph. The domain is all real numbers.

The $y$ values that appear start at $3$ and go up. We get all numbers greater than or equal to $3$. Inequality: $y \ge 3$.
If you've learned interval notation, you write: $\left[3 , \infty\right)$

${x}^{2} + {y}^{2} / 9 \le 1$

 graph{x^2 + y^2/9 <= 1 [-5.35, 7.14, -3.105, 3.14]}

Domain (x-values) Go from $- 1$ to $1$ (inequality: $- 1 \le x \le 1$)(interval: $\left[- 1 , 1\right]$)

Range: $- 3$ to $3$ (inequal: $- 3 \le y \le 3$(interval: $\left[- 3 , 3\right]$)

More challenging:

${x}^{2} + {y}^{2} < 9$

graph{x^2+y^2 < 9 [-5.35, 7.14, -3.105, 3.14]}

The dotted line is not included, so we do not include the points at $- 3$ or at $3$.

Domain: $- 3 < x < 3$ (interval (-3, 3)) Range: -3<y<3 (interval #(-3, 3))

Last one:

$x - {y}^{2} < 6$

graph{x- y^2 < 6 [-9.87, 30.68, -9.96, 10.32]}

Domain: all real numbers
Range: all real numbers