Relating Domain to Context and Graphs - Algebra 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

8. Which axis corresponds to the domain when reading a function graph?

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

9. What domain is represented by a graph drawn only for $x\le -1$?

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

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

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

11. Which domain best matches a function $s(n)$ giving the $n$th term of a sequence?

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

12. What is the domain of $f(x)=\sqrt{x}$ over the real numbers?

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

13. What is the domain of a function in the context of inputs and outputs?

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

14. What is the domain shown on a graph for a function $y=f(x)$?

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

15. Which domain type fits a function modeling time in seconds after an event starts?

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

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

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

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

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

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

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

19. What is the domain of $f(x)=|x-7|$ over the real numbers?

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

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

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

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

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

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

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

23. Which domain is appropriate for $T(n)$ = total points after $n$ games played?

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

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

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

25. Which domain best matches $m(a)$ = monthly payment depending on APR $a$ (in decimal)?

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

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

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

27. Which domain is appropriate for $c(n)$ = cost to ship $n$ boxes?

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

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

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

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

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

30. What is an appropriate domain for $V(t)$ = volume of water after $t$ minutes of filling?

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

31. What is the domain of the linear function $f(x)=2x-5$ over the reals?

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

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

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

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

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

34. What domain is appropriate for $h(n)$ = person-hours to assemble $n$ engines?

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

35. Which domain best matches a function $p(w)$ giving price for $w$ pounds of fruit?

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

36. What does an open circle at $x=4$ indicate about the domain endpoint?

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

37. What does a closed dot at $x=4$ indicate about the domain endpoint?

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

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

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

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

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

40. What domain is represented by a graph drawn only for $-2\le x<5$?

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

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

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

42. Which interval notation matches the domain $x\ge -4$?

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

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

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

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

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

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

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

46. Which domain is appropriate for $b(t)$ = bank balance $t$ days after opening an account?

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

47. What is a common domain restriction for a real-world function counting objects?

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

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

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

Study these flashcards