Algebra 2 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 2.

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: Identify the domain restriction for a square root \sqrt{x-2}.

Answer: Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.

Flashcard 2: What is the domain of $f(x)=\sqrt{5-x}$?

Answer: $x\le 5$, so $(-\infty,5]$. The radicand $(5-x)$ must be non-negative.

Flashcard 3: What is the domain of a function in terms of allowable inputs?

Answer: The set of all allowable $x$-values (inputs). The domain defines which $x$-values are valid inputs for the function.

Flashcard 4: What does it mean if a graph has a point at $x=3$?

Answer: $3$ is included in the domain. A plotted point shows the function is defined at that $x$-value.

Flashcard 5: What does an open circle at $x=3$ on a graph indicate about the domain?

Answer: $3$ is excluded from the domain. Open circles indicate undefined points, creating domain restrictions.

Flashcard 6: Identify the domain restriction caused by a denominator of $x-5$.

Answer: Exclude $x=5$ from the domain. Denominators equal zero create undefined points.

Flashcard 7: Identify the domain restriction for a square root \sqrt{x-2}.

Answer: Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.

Flashcard 8: Identify the domain restriction for an even root $\sqrt{3-2x}$.

Answer: Require $3-2x\ge 0$, so $x\le \frac{3}{2}$. Even roots need non-negative radicands, so solve the inequality.

Flashcard 9: What is the typical domain of a polynomial function $f(x)$?

Answer: All real numbers, $(-\infty,\infty)$. Polynomials are defined for all real number inputs.

Flashcard 10: What is the typical domain of an exponential function $f(x)=a\cdot b^x$?

Answer: All real numbers, $(-\infty,\infty)$. Exponential functions accept any real number exponent.

Flashcard 11: What is the typical domain of a logarithmic function $f(x)=\log(x)$?

Answer: Require $x>0$. Logarithms require positive arguments to be defined.

Flashcard 12: What domain restriction is created by $f(x)=\log(x-4)$?

Answer: Require $x-4>0$, so $x>4$. The logarithm argument $(x-4)$ must be positive.

Flashcard 13: What is the domain of $f(x)=\frac{1}{x^2+1}$?

Answer: All real numbers, $(-\infty,\infty)$. The denominator $x^2+1$ is never zero for real $x$.

Flashcard 14: What is the domain of $f(x)=\frac{1}{x-7}$?

Answer: All real numbers except $x=7$. The denominator equals zero when $x=7$, creating a restriction.

Flashcard 15: What is the domain of $f(x)=\sqrt{x}$?

Answer: $x\ge 0$, so $[0,\infty)$. Square roots require non-negative radicands.

Flashcard 16: What is the domain of $f(x)=\log(x+2)$?

Answer: $x>-2$, so $(-2,\infty)$. The logarithm argument $(x+2)$ must be positive.

Flashcard 17: What is the domain of $f(x)=\frac{x+1}{x^2-9}$?

Answer: All real numbers except $x=\pm 3$. Factor $x^2-9=(x-3)(x+3)$ to find where denominator equals zero.

Flashcard 18: What is the domain of $f(x)=\frac{1}{(x-2)(x+5)}$?

Answer: All real numbers except $x=2$ and $x=-5$. Each factor in the denominator creates a domain restriction.

Flashcard 19: What is the domain of $f(x)=\frac{1}{\sqrt{x-3}}$?

Answer: Require $x-3>0$, so $(3,\infty)$. Square root in denominator requires strictly positive radicand.

Flashcard 20: What is the domain of $f(x)=\log(2-x)$?

Answer: Require $2-x>0$, so $(-\infty,2)$. Logarithm argument $(2-x)$ must be positive.

Flashcard 21: What is the domain of $f(x)=\frac{\sqrt{x-4}}{x-4}$?

Answer: Require $x>4$, so $(4,\infty)$. Both numerator and denominator require $x>4$.

Flashcard 22: What is the domain of $f(x)=\sqrt{x^2-9}$?

Answer: $x\le -3$ or $x\ge 3$. Solve $x^2-9\ge 0$ using factoring and sign analysis.

Flashcard 23: What is the domain of $f(x)=\frac{1}{x^2-4x+4}$?

Answer: All real numbers except $x=2$. The denominator $(x-2)^2$ equals zero only when $x=2$.

Flashcard 24: What is the domain of $f(x)=|x-5|$?

Answer: All real numbers, $(-\infty,\infty)$. Absolute value functions are defined for all real numbers.

Flashcard 25: Which domain is appropriate for $h(n)$ = engines assembled when $n$ is a count?

Answer: Positive integers, $n\in\{1,2,3,\dots\}$. Engine counts must be positive whole numbers.

Flashcard 26: Which domain is appropriate for $A(r)=\pi r^2$ when $r$ is a radius?

Answer: Nonnegative real numbers, $r\ge 0$. Radius measurements cannot be negative in real contexts.

Flashcard 27: Which domain is appropriate for $t$ in a model describing time after a start?

Answer: Nonnegative real numbers, $t\ge 0$. Time after a starting point cannot be negative.

Flashcard 28: Which domain is appropriate for $d(t)$ distance traveled after $t$ seconds?

Answer: Nonnegative real numbers, $t\ge 0$. Distance traveled cannot be negative in standard contexts.

Flashcard 29: Which domain is appropriate for $P(n)$ population after $n$ years (counted in years)?

Answer: Nonnegative integers, $n\in\{0,1,2,\dots\}$. Years counted as whole number increments from zero.

Flashcard 30: Which domain is appropriate for $C(x)$ cost to buy $x$ pounds of fruit?

Answer: Nonnegative real numbers, $x\ge 0$. Weight purchases cannot be negative quantities.