Domain and Range for MATH 139

What is Domain and Range?

Every function is like a machine: you put something in (the input), and you get something out (the output). But not every machine accepts every input — and not every output is possible.

• Domain = all the valid inputs (x-values) you can plug into the function
• Range = all the possible outputs (y-values) the function can produce

Understanding domain and range helps you know where a function "lives" — which values make sense to use and which results you can expect.

Visualizing Domain and Range

In this graph, notice that:

• Domain: You can plug in ANY x-value (the parabola extends left and right forever) → Domain: $(-\infty, \infty)$
• Range: The outputs only go from 0 upward (the parabola never dips below the x-axis) → Range: $[0, \infty)$

Interval Notation Quick Reference

💡 Key rule: Always use parentheses with $\infty$ or $-\infty$ — you can never "reach" infinity, so you can't include it!

Finding the Domain

The domain is usually "all real numbers" except where the function breaks. Look for these common restrictions:

Restriction 1: Division by Zero

You cannot divide by zero. If the function has a denominator, set it $\neq 0$.

$$f(x) = \frac{1}{x-3}$$

The denominator is $x - 3$. Set it $\neq 0$:

$$x - 3 \neq 0 \implies x \neq 3$$

Domain: $(-\infty, 3) \cup (3, \infty)$ — all real numbers except 3.

Restriction 2: Square Roots of Negatives

You cannot take the square root of a negative number (in real numbers). The expression under the root must be $\geq 0$.

$$g(x) = \sqrt{x + 2}$$

Set the inside $\geq 0$:

$$x + 2 \geq 0 \implies x \geq -2$$

Domain: $[-2, \infty)$

Restriction 3: Logarithms of Non-Positives

You cannot take the logarithm of zero or a negative number. The input to a log must be $> 0$.

$$h(x) = \ln(x - 5)$$

Set the inside $> 0$:

$$x - 5 > 0 \implies x > 5$$

Domain: $(5, \infty)$

Combining Restrictions

Some functions have multiple restrictions. Find each one, then combine.

$$f(x) = \frac{\sqrt{x}}{x - 4}$$

• Square root: $x \geq 0$
• Denominator: $x \neq 4$

Domain: $[0, 4) \cup (4, \infty)$

Finding the Range

The range is often trickier than the domain. Here are strategies:

Strategy 1: Use the Graph

For $f(x) = \frac{1}{x}$:

• The graph approaches but never touches $y = 0$
• The function produces every other y-value
• Range: $(-\infty, 0) \cup (0, \infty)$ — all real numbers except 0

Strategy 2: Think About What Outputs Are Possible

For $f(x) = x^2$:

• Squaring any number gives a non-negative result
• You can get 0 (when $x = 0$)
• You can get arbitrarily large values (as $x \to \pm\infty$)
• Range: $[0, \infty)$

Strategy 3: Solve for x in Terms of y

Set $y = f(x)$ and solve for $x$. The values of $y$ that give valid solutions are in the range.

For $f(x) = 2x + 1$:

$$y = 2x + 1$$ $$y - 1 = 2x$$ $$x = \frac{y - 1}{2}$$

This gives a valid $x$ for ANY value of $y$. Range: $(-\infty, \infty)$

Example 3: Piecewise Function from a Graph

Using the graph below, find the domain and range of the piecewise function.

Show Solution

When reading domain and range from a graph:

• Domain: Look at how far left and right the graph extends (x-axis coverage)
• Range: Look at how far up and down the graph extends (y-axis coverage)
• Closed dots (●) mean the point IS included — use a bracket $[$
• Open dots (○) mean the point is NOT included — use a parenthesis $($

Finding the Domain:

• The first piece extends from the upper left (toward $x = -\infty$) and ends at $x = 0$ with an open circle
• The second piece starts at $x = 0$ with a closed dot and ends at $x = 3.5$ with an open circle
• The third piece starts at $x = 3.5$ with an open circle and extends to the lower right (toward $x = +\infty$)

At $x = 0$: The closed dot means $x = 0$ IS included.

At $x = 3.5$: Both adjacent pieces have open circles, so $x = 3.5$ is NOT included.

$$\boxed{\text{Domain: } (-\infty, 3.5) \cup (3.5, \infty)}$$

Finding the Range:

• The first piece has an arrow pointing up toward $+\infty$
• The third piece has an arrow pointing down toward $-\infty$
• Together, every y-value from $-\infty$ to $+\infty$ is covered

$$\boxed{\text{Range: } (-\infty, \infty)}$$

Common Mistakes and Misunderstandings

❌ Mistake: Using brackets with infinity

Wrong: $[0, \infty]$

Why it's wrong: Infinity is not a number — you can never "reach" it. You can't include something that doesn't exist as a specific value.

Correct: $[0, \infty)$

❌ Mistake: Forgetting to check ALL restrictions

Wrong: For $f(x) = \frac{\sqrt{x}}{x-1}$, saying domain is $x \neq 1$

Why it's wrong: You found the denominator restriction but forgot the square root restriction ($x \geq 0$).

Correct: Domain: $[0, 1) \cup (1, \infty)$

❌ Mistake: Confusing domain and range

Wrong: "The range of $f(x) = x^2$ is all real numbers because you can plug in any x."

Why it's wrong: That describes the domain, not the range. The range asks what outputs are possible.

Correct: Domain: $(-\infty, \infty)$. Range: $[0, \infty)$ because $x^2$ is never negative.