Finding Domain from a Graph

Find the domain of the function shown, where the graph exists from $x = -3$ to $x = 5$.

Identify the leftmost x-value: The graph starts at $x = -3$ = Left bound: $-3$

Identify the rightmost x-value: The graph ends at $x = 5$ = Right bound: $5$

Write in interval notation: All x-values from $-3$ to $5$, inclusive = $[-3, 5]$

Write in set notation (alternative): $\{x \mid -3 \leq x \leq 5\}$ = "x such that x is between -3 and 5"

Answer: Domain: $[-3, 5]$ or $\{x \mid -3 \leq x \leq 5\}$

Finding Range from a Graph

Find the range of a parabola with vertex at $(0, -4)$ that opens upward.

Find the minimum y-value: The vertex is the lowest point at $y = -4$ = Minimum: $y = -4$

Determine the maximum y-value: The parabola goes up forever = Maximum: $\infty$

Write in interval notation: All y-values from $-4$ to infinity = $[-4, \infty)$

Interpret: The function outputs all values $\geq -4$ = Range includes $-4$ and all greater values

Answer: Range: $[-4, \infty)$ or $\{y \mid y \geq -4\}$

Domain with Restrictions

Find the domain of $f(x) = \frac{1}{x-2}$

Identify restrictions: A fraction is undefined when the denominator equals zero = Need: $x - 2 \neq 0$

Solve for restricted value: $x - 2 = 0 \Rightarrow x = 2$ = $x \neq 2$

Write the domain: All real numbers except 2 = $(-\infty, 2) \cup (2, \infty)$

Verify: At $x = 2$: $f(2) = \frac{1}{0}$ is undefined = Restriction confirmed

Answer: Domain: $(-\infty, 2) \cup (2, \infty)$ or $\{x \mid x \neq 2\}$

Domain of Square Root Function

Find the domain of $g(x) = \sqrt{x + 3}$

Recall the restriction: The expression under a square root must be $\geq 0$ = Need: $x + 3 \geq 0$

Solve the inequality: $x + 3 \geq 0 \Rightarrow x \geq -3$ = $x \geq -3$

Write in interval notation: All x-values from $-3$ to infinity = $[-3, \infty)$

Check boundary: At $x = -3$: $\sqrt{-3+3} = \sqrt{0} = 0$ (valid) = $-3$ is included

Answer: Domain: $[-3, \infty)$ or $\{x \mid x \geq -3\}$

Domain and Range from Equation

Find the domain and range of $f(x) = x^2 - 1$

Check for domain restrictions: No fractions (no division by zero) and no square roots (no negative under root) = No restrictions

State the domain: Any real number can be squared = Domain: $(-\infty, \infty)$

Find the minimum output: Since $x^2 \geq 0$ for all x, the minimum of $x^2 - 1$ is $0 - 1 = -1$ = Minimum y-value: $-1$

State the range: The parabola opens up from $y = -1$ = Range: $[-1, \infty)$

Answer: Domain: $(-\infty, \infty)$, Range: $[-1, \infty)$

Mistake: Confusing domain with range

Why: Both terms describe sets of values, but they apply to different variables.

Correct: Domain = x-values (horizontal, inputs). Range = y-values (vertical, outputs). Remember: "D" comes before "R" alphabetically, just like x comes before y.

Mistake: Forgetting to exclude values that make denominators zero

Why: Division by zero is undefined, so those x-values cannot be in the domain.

Correct: Always check if there's a variable in the denominator. Set denominator $\neq 0$ and solve.

Mistake: Using wrong bracket notation

Why: Square brackets $[ ]$ include endpoints; parentheses $( )$ exclude them.

Correct: $[a, b]$ means $a \leq x \leq b$ (includes a and b). $(a, b)$ means $a < x < b$ (excludes a and b). Always use $( )$ with $\infty$.

Mistake: Assuming all functions have domain $(-\infty, \infty)$

Why: Many functions have natural restrictions from fractions, square roots, or real-world contexts.

Correct: Check for: division by zero, square roots of negatives, logarithms of non-positives, and contextual limits.

Projectile Motion

When a ball is thrown, its height over time is a function with restricted domain and range.

If $h(t) = -5t^2 + 20t$ models height in meters after t seconds, the domain is $[0, 4]$ (time from launch to landing) and range is $[0, 20]$ (ground level to maximum height).

Pricing Functions

Businesses use functions to model prices, with domains restricted to realistic quantities.

If $P(x) = 100 - 2x$ gives the price per unit when selling x units, the domain might be $[0, 50]$ (can't sell negative units, and price can't go below zero).

Q: How do I know if an endpoint should use a bracket or parenthesis?

A: Use a bracket $[ ]$ if the endpoint is included in the set (solid dot on graph, $\leq$ or $\geq$ in inequality). Use parenthesis $( )$ if excluded (open dot, $<$ or $>$). Always use parentheses with infinity since infinity is not a number.

Q: What if a function has no restrictions?

A: Then the domain is all real numbers: $(-\infty, \infty)$ or $\mathbb{R}$. This is common for polynomials without fractions or roots, like $f(x) = 2x + 3$ or $g(x) = x^3 - x$.

Q: How do I find the range without graphing?

A: Consider what outputs are possible. For quadratics, find the vertex (minimum or maximum). For square roots, outputs are always $\geq 0$. For rational functions, check horizontal asymptotes. Sometimes graphing is the easiest method!