Relating Domain to Context and Graphs - Algebra 2

1. What does a vertical asymptote at $x=a$ imply about the domain?

   $x=a$ is excluded from the domain. Vertical asymptotes occur where denominators equal zero.

2. Which domain is appropriate for $h(n)$ = engines assembled when $n$ is a count?

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

3. What is the domain of $f(x)=\frac{1}{x-7}$?

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

4. What is the domain of a function in terms of allowable inputs?

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

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

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

6. Which domain is appropriate for $A(r)=\pi r^2$ when $r$ is a radius?

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

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

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

8. What is the domain of $f(x)=\frac{1}{x}$ based on its graph behavior?

   All real numbers except $x=0$. Vertical asymptote at $x=0$ excludes this value.

9. Identify the domain for $n$ in $h(n)$ person-hours to assemble $n$ engines.

   Positive integers, $n\in\{1,2,3,\dots\}$. Engine assembly requires at least one engine to be meaningful.

10. What does a right endpoint at $x=5$ with an open dot indicate for the domain?

    The domain goes up to but does not include $x=5$. Open dots exclude the endpoint from the domain.

11. What is the domain of $f(x)=\log(x+2)$?

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

12. Find the appropriate domain for $V(t)$ voltage measured for $0\le t\le 10$ seconds.

    $0\le t\le 10$. Measurement context limits the time interval.

13. Which domain is most appropriate for $T(x)$ temperature at depth $x$ meters below surface?

    Nonnegative real numbers, $x\ge 0$. Depth measurements cannot be negative below surface.

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

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

15. Which domain is appropriate for $m$ = number of tickets sold?

    Nonnegative integers, $m\in\{0,1,2,\dots\}$. Ticket counts must be whole numbers starting from zero.

16. What is the domain of $f(x)=\sqrt{x-9}$ based on its left endpoint?

    All real $x$ with $x\ge 9$. Left endpoint shows where the function begins.

17. What is the domain of $f(x)=\sqrt{5-x}$?

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

18. What does it mean if a graph has a point at $x=3$?

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

19. What domain restriction is created by $f(x)=\log(x-4)$?

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

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

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

21. Find the appropriate domain for height $h(t)$ of a ball for $t$ seconds after launch.

    $t\ge 0$ until the ball hits the ground. Physical context limits time until impact occurs.

22. Which domain is appropriate for $P(n)$ population after $n$ years (counted in years)?

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

23. What does a left endpoint at $x=2$ with a closed dot indicate for the domain?

    The domain includes $x=2$ and extends rightward. Closed dots include the endpoint in the domain.

24. Which domain is most appropriate for $f(x)$ where $x$ is a persons age in years?

    Nonnegative real numbers, $x\ge 0$. Age cannot be negative in real-world contexts.

25. Identify the domain of $f(x)=\frac{x^2-1}{x-1}$ including any hole.

    All real numbers except $x=1$. Factor cancellation creates a hole, not domain inclusion.

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

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

27. Which domain is appropriate for $C(x)$ cost to buy $x$ pounds of fruit?

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

28. What is the typical domain of a logarithmic function $f(x)=\log(x)$?

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

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

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

30. What is the domain of $f(x)=\frac{\sqrt{x-4}}{x-4}$?

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

31. What does an open circle at $x=3$ on a graph indicate about the domain?

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

32. What is the domain of $f(x)=\sqrt{x}$?

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

33. What does a gap (hole) in a graph at $x=a$ imply about the domain?

    $x=a$ is not in the domain. Gaps indicate missing points where function is undefined.

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

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

35. What is the domain of $f(x)=\log(2-x)$?

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

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

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

37. Identify the domain restriction caused by a denominator of $x-5$.

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

38. Which domain is appropriate for $d(t)$ distance traveled after $t$ seconds?

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

39. Identify the domain in words for $[2,7)$.

    All real $x$ with $2\le x<7$. Bracket notation shows inclusion and exclusion of endpoints.

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

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

41. Identify the domain from an interval notation $(-\infty,3)\cup(3,\infty)$.

    All real numbers except $x=3$. Union notation shows all values except the excluded point.

42. Which domain is most appropriate for $s(n)$ = salary after $n$ full years at a job?

    Nonnegative integers, $n\in\{0,1,2,\dots\}$. Full years of employment are counted as integers.

43. What is the typical domain of a polynomial function $f(x)$?

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

44. What does a graph drawn only for $x\ge 0$ imply about the domain?

    The domain is restricted to $x\ge 0$. Graph bounds show the function's applicable input range.

45. What is the domain of $f(x)=|x-5|$?

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

46. Identify the domain of $f(x)=\frac{x}{x(x-1)}$ after simplifying the expression.

    All real numbers except $x=0$ and $x=1$. Original restrictions remain even after algebraic simplification.

47. What is the domain of $f(x)=\log(x)$ based on its vertical asymptote?

    All real $x$ with $x>0$. Vertical asymptote at $x=0$ restricts the domain.

48. Which domain is appropriate for $t$ in a model describing time after a start?

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

49. What is the domain of $f(x)=\sqrt{x+9}$?

    $x\ge -9$, so $[-9,\infty)$. The radicand $(x+9)$ must be non-negative.

50. Which domain is appropriate for $x$ in $p(x)$ price as a function of quantity $x$?

    Typically $x\ge 0$; often integers if items are countable. Context determines if quantities are continuous or discrete.

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