Algebra Flashcards: Relating Domain To Context And Graphs

What this deck covers

This deck focuses on Relating Domain To Context And Graphs, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the domain of $f(x)=\frac{1}{\sqrt{x}}$ over the real numbers?

Answer: $x>0$. Need $x>0$ for both denominator and square root.

Flashcard 2: What is the domain of $f(x)=\frac{\sqrt{x+2}}{x-4}$ over the real numbers?

Answer: $x\ge -2$ and $x\ne 4$. Need $x\ge -2$ for square root and $x\ne 4$ for denominator.

Flashcard 3: Which interval notation matches the domain $x<1$ or $x\ge 5$?

Answer: $(-\infty,1)\cup[5,\infty)$. Union combines two separate intervals.

Flashcard 4: What domain should be used for $C(d)=3.50d$ if $d$ is distance traveled in miles?

Answer: All real $d\ge 0$. Distance cannot be negative in real-world contexts.

Flashcard 5: What domain should be used for $P(n)=12n$ if $n$ is the number of items purchased?

Answer: Nonnegative integers $n=0,1,2,\dots$. Can purchase zero or any whole number of items.

Flashcard 6: What domain is represented by a graph drawn only for $x>3$?

Answer: $(3,\infty)$. Parenthesis excludes $3$, infinity is always open.

Flashcard 7: Which interval notation matches the domain $-6<x\le 2$?

Answer: $(-6,2]$. Parenthesis excludes $-6$, bracket includes $2$.

Flashcard 8: What domain is represented by a graph drawn only for $x>3$?

Answer: $(3,\infty)$. Parenthesis excludes $3$, infinity is always open.

Flashcard 9: Which axis corresponds to the domain when reading a function graph?

Answer: The $x$-axis. Domain values are plotted horizontally on the $x$-axis.

Flashcard 10: What domain is represented by a graph drawn only for $x\le -1$?

Answer: $(-\infty,-1]$. Infinity is open, bracket includes $-1$.

Flashcard 11: What is the domain of $f(x)=\frac{1}{\sqrt{x-7}}$ over the real numbers?

Answer: $x>7$. Need $x-7>0$ for the square root in denominator.

Flashcard 12: Which domain best matches a function $s(n)$ giving the $n$th term of a sequence?

Answer: Positive integers $n=1,2,3,\dots$. Sequence terms are indexed by positive integers.

Flashcard 13: What is the domain of $f(x)=\sqrt{x}$ over the real numbers?

Answer: $x\ge 0$. Square root requires non-negative radicand.

Flashcard 14: What is the domain of a function in the context of inputs and outputs?

Answer: The set of all allowable input values for the function. Domain consists of valid inputs that produce defined outputs.

Flashcard 15: What is the domain shown on a graph for a function $y=f(x)$?

Answer: All $x$-values for which the graph has at least one point. Domain is the horizontal extent of the graph.

Flashcard 16: Which domain type fits a function modeling time in seconds after an event starts?

Answer: All real numbers $t\ge 0$. Time cannot be negative in this context.

Flashcard 17: What is the domain of $f(x)=\frac{3}{x-2}$ over the real numbers?

Answer: All real $x$ except $x=2$. Denominator cannot equal zero, so $x\ne 2$.

Flashcard 18: What is the domain of $f(x)=\frac{1}{x(x-5)}$ over the real numbers?

Answer: All real $x$ except $x=0$ and $x=5$. Denominator $x(x-5)$ cannot equal zero.

Flashcard 19: What domain should be used for $A(r)=\pi r^2$ when $r$ is a radius?

Answer: All real $r\ge 0$. Radius must be non-negative in geometric contexts.

Flashcard 20: What is the domain of $f(x)=|x-7|$ over the real numbers?

Answer: All real numbers. Absolute value is defined for all real numbers.

Flashcard 21: What is the domain of a function whose graph begins at $x=0$ with a closed dot and continues right?

Answer: $[0,\infty)$. Closed dot includes the starting point $x=0$.

Flashcard 22: What is the domain of a function whose graph begins at $x=0$ with an open circle and continues right?

Answer: $(0,\infty)$. Open circle excludes the starting point $x=0$.

Flashcard 23: What is the domain of $f(x)=\frac{1}{\sqrt{x}}$ over the real numbers?

Answer: $x>0$. Need $x>0$ for both denominator and square root.

Flashcard 24: What is the domain of $f(x)=\frac{1}{x^2+4}$ over the real numbers?

Answer: All real numbers. Denominator $x^2+4$ is always positive for real $x$.

Flashcard 25: Which domain is appropriate for $T(n)$ = total points after $n$ games played?

Answer: Nonnegative integers $n=0,1,2,\dots$. Can have played zero or any whole number of games.

Flashcard 26: What is the domain of $f(x)=\sqrt{(x-4)^2}$ over the real numbers?

Answer: All real numbers. $(x-4)^2$ is always non-negative, so square root exists.

Flashcard 27: What is the domain of $f(x)=\sqrt{9-x^2}$ over the real numbers?

Answer: $-3 \le x \le 3$. Need $9-x^2 \ge 0$, so $x^2 \le 9$.

Flashcard 28: Which domain best matches $m(a)$ = monthly payment depending on APR $a$ (in decimal)?

Answer: All real $a>0$ (in the realistic range for APR). APR must be positive for realistic loan calculations.

Flashcard 29: What is the domain of $f(x)=\frac{x}{x}$ when viewed as a rational expression?

Answer: All real $x$ except $x=0$. Cannot divide by zero even though $\frac{x}{x}=1$ when $x\ne 0$.

Flashcard 30: Which domain is appropriate for $c(n)$ = cost to ship $n$ boxes?

Answer: Positive integers $n=1,2,3,\dots$. Must ship at least one box.

Flashcard 31: What domain is represented by a graph drawn only for $x\le -1$?

Answer: $(-\infty,-1]$. Infinity is open, bracket includes $-1$.

Flashcard 32: Which inequality describes the domain of $f(x)=\sqrt{5-2x}$ over the reals?

Answer: $x\le \frac{5}{2}$. Need $5-2x\ge 0$ for the square root.

Flashcard 33: What domain should be used for $g(t)$ = height of a ball $t$ seconds after it is thrown?

Answer: All real $t\ge 0$ (often until the ball hits the ground). Time starts at $0$ and continues until impact.

Flashcard 34: What is an appropriate domain for $V(t)$ = volume of water after $t$ minutes of filling?

Answer: All real $t\ge 0$ (often until the container is full). Time starts at $0$ and continues until full.

Flashcard 35: What is the domain of the linear function $f(x)=2x-5$ over the reals?

Answer: All real numbers. Linear functions are defined for all real numbers.

Flashcard 36: What is the domain of $f(x)=\frac{x+1}{x^2-9}$ over the real numbers?

Answer: All real $x$ except $x=-3$ and $x=3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.

Flashcard 37: Identify the domain of $f(x)=\frac{2}{(x+1)^2}$ over the real numbers.

Answer: All real $x$ except $x=-1$. Denominator $(x+1)^2$ cannot equal zero.

Flashcard 38: What domain is appropriate for $h(n)$ = person-hours to assemble $n$ engines?

Answer: Positive integers $n=1,2,3,\dots$. Cannot assemble a fractional number of engines.

Flashcard 39: Which domain best matches a function $p(w)$ giving price for $w$ pounds of fruit?

Answer: All real $w\ge 0$. Weight must be non-negative for pricing.

Flashcard 40: What does an open circle at $x=4$ indicate about the domain endpoint?

Answer: The value $x=4$ is not included in the domain. Open circle means the endpoint is excluded.

Flashcard 41: What does a closed dot at $x=4$ indicate about the domain endpoint?

Answer: The value $x=4$ is included in the domain. Closed dot means the endpoint is included.

Flashcard 42: What is the domain of $f(x)=\frac{x^2-1}{x-1}$ when viewed as a rational expression?

Answer: All real $x$ except $x=1$. Denominator cannot be zero even though it simplifies to $x+1$.

Flashcard 43: What is the domain of $f(x)=\frac{1}{x}$ over the real numbers?

Answer: All real $x$ except $x=0$. Cannot divide by zero in the denominator.

Flashcard 44: What domain is represented by a graph drawn only for $-2\le x<5$?

Answer: $[-2,5)$. Bracket includes $-2$, parenthesis excludes $5$.

Flashcard 45: What is the domain of a function whose graph exists for all $x$ except a hole at $x=2$?

Answer: All real $x$ except $x=2$. Hole means that specific $x$-value is excluded.

Flashcard 46: Which interval notation matches the domain $x\ge -4$?

Answer: $[-4,\infty)$. Bracket means $-4$ is included, infinity is always open.

Flashcard 47: What is the domain of $f(x)=\sqrt{x-5}$ over the real numbers?

Answer: $x\ge 5$. Need $x-5\ge 0$ for the square root to be real.

Flashcard 48: What is the domain of the quadratic function $f(x)=x^2-4x+1$ over the reals?

Answer: All real numbers. Quadratic functions are defined for all real numbers.

Flashcard 49: What is the domain of $f(x)=\sqrt{2x+1}$ over the real numbers?

Answer: $x\ge -\frac{1}{2}$. Need $2x+1\ge 0$ for the square root.

Flashcard 50: Which domain is appropriate for $b(t)$ = bank balance $t$ days after opening an account?

Answer: All real $t\ge 0$ (often integers if measured by whole days). Time cannot be negative, may be continuous or discrete.

Flashcard 51: What is a common domain restriction for a real-world function counting objects?

Answer: Nonnegative integers (or positive integers if $0$ is not allowed). Counting requires whole numbers, starting from $0$ or $1$.

Flashcard 52: What is the domain of $f(x)=\sqrt{9-x^2}$ over the real numbers?

Answer: $-3\le x\le 3$. Need $9-x^2\ge 0$, so $x^2\le 9$.