About Domain and Range:

When working with relations and functions, we will sometimes be asked to find the domain and range of a relation or function from its equation or its graph. If we want to find the domain, we will inspect our equation and ask the question: what is allowed as a replacement for our independent variable x? For the range, we will think about the possible outputs or y-values, given the possible x-values.

Test Objectives
  • Demonstrate the ability to find the domain and range of a relation or function from a graph
  • Demonstrate the ability to find the domain of a relation or function
Domain and Range Practice Test:

#1:

Instructions: find the domain and range.

$$a)\hspace{.2em}y=-3x - 2$$ graph of y=-3x - 2

$$b)\hspace{.2em}y=\sqrt{x + 1}- 3$$ graph of y=sqrt(x + 1) - 3

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#2:

Instructions: find the domain.

$$a)\hspace{.2em}y=\frac{2x}{|3x - 3| - 6}$$

$$b)\hspace{.2em}y=\frac{5x - 3}{\sqrt{\left|x^2 + x - 2\right| - 4}}$$

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#3:

Instructions: find the domain.

$$a)\hspace{.2em}y=-\frac{7}{x^2 - 4x - 5}$$

$$b)\hspace{.2em}y=-\frac{x - 2}{x^2 - 5x + 6}$$

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#4:

Instructions: find the domain.

$$a)\hspace{.2em}y=\frac{\sqrt{x + 4}}{2x - 6}$$

$$b)\hspace{.2em}y=\frac{\sqrt{3x - 2}}{6x^2 + 8x - 30}$$

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#5:

Instructions: find the domain.

$$a)\hspace{.2em}y=\frac{\sqrt{6x^2 + 11x - 7}}{\sqrt{\left|2x - 5\right| + 4x^2 - 2x - 13}}$$

$$b)\hspace{.2em}y=\frac{\sqrt{35x^2 - 58x - 9}}{\sqrt{x^2 - 2x + 1}}$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\text{Domain:}\hspace{.2em}\{x|x \in \mathbb{R}\}$$ $$\text{Range:}\hspace{.2em}\{y|y \in \mathbb{R}\}$$

$$b)\hspace{.2em}\text{Domain:}\hspace{.2em}\{x|x≥ -1\}$$ $$\text{Range:}\hspace{.2em}\{y|y ≥ -3\}$$

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#2:

Solutions:

Note: The graphs below have been produced with the help of ASCIIsvg.

$$a)\hspace{.2em}\hspace{.2em}\text{Domain:}\hspace{.2em}\{x|x \ne -1, 3\}$$ graphing y=(2x)/(|3x - 3| - 6)

$$b)\hspace{.2em}\hspace{.2em}\text{Domain:}\hspace{.2em}\{x|x < -3, x > 2\}$$ graphing y=(5x - 3)/(sqrt(|x^2 + x - 2| - 4))

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#3:

Solutions:

Note: The graphs below have been produced with the help of ASCIIsvg.

$$a)\hspace{.2em}\text{Domain:}\hspace{.2em}\{x | x ≠ -1, 5\}$$ graphing y=(-7)/(x^2 - 4x - 5)

$$b)\hspace{.2em}\text{Domain:}\hspace{.2em}\{x | x ≠ 2, 3\}$$ graphing y=-(x - 2)/(x^2 - 5x + 6)

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#4:

Solutions:

Note: The graphs below have been produced with the help of ASCIIsvg.

$$a)\hspace{.2em}\text{Domain:}\hspace{.2em}\left\{x | x ≥ -4, x ≠ 3\right\}$$ graphing y=(sqrt(x + 4))/(2x - 6)

$$b)\hspace{.2em}\text{Domain:}\hspace{.2em}\left\{x | x ≥ \frac{2}{3}, x ≠ \frac{5}{3}\right\}$$ graphing y=(sqrt(3x - 2))/(6x^2 + 8x - 30)

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#5:

Solutions:

Note: The graphs below have been produced with the help of ASCIIsvg.

$$a)\hspace{.2em}\text{Domain:}\hspace{.2em}\left\{x | x ≤ -\frac{7}{3}, x > 2\right\}$$ graphing y=(sqrt(6x^2 + 11x - 7))/(sqrt(|2x - 5| + 4x^2 - 2x - 13))

$$b)\hspace{.2em}\text{Domain:}\hspace{.2em}\left\{x | x ≤ -\frac{1}{7}, x ≥ \frac{9}{5}\right\}$$ graphing y=(sqrt(35x^2 - 58x - 9))/(sqrt(x^2 - 2x + 1))

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