Restrict Domain to Make Invertible - Algebra 2

1. What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\ge 0$?

   $f^{-1}(x)=\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the positive square root.

2. What is the vertex $x$-value used to split a parabola $f(x)=a(x-h)^2+k$ into one-to-one halves?

   $x=h$. The vertex divides the parabola into symmetric halves.

3. What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\le 0$?

   $f^{-1}(x)=-\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the negative square root.

4. What is the range of $f(x)=-(x+2)^2$ when the domain is restricted to $x\ge -2$?

   Range is $y\le 0$. The parabola opens downward with vertex at $(-2,0)$.

5. Which domain restriction makes $f(x)=x^2$ invertible and uses the negative branch?

   Restrict to $x\le 0$. This restriction uses the left branch of the parabola.

6. What condition must a function satisfy to have an inverse function (as a function)?

   $f$ must be one-to-one on its domain. Without this condition, multiple inputs would map to the same output.

7. What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\ge 3$?

   $f^{-1}(x)=3+\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the positive square root.

8. What restriction makes $f(x)=(x-3)^2$ invertible to match the negative square root branch?

   Restrict to $x\le 3$. The vertex is at $x=3$, so restrict to the left side.

9. What restriction makes $f(x)=x^2$ one-to-one on an interval centered at $0$?

   Use $x\ge 0$ or $x\le 0$ (one side only). Split at the vertex to create monotonic intervals.

10. What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\le 1$?

    $f^{-1}(x)=1-\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the left branch.

11. What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\le -2$?

    $f^{-1}(x)=-2-\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the negative branch.

12. Identify the correct inverse for restricted $f(x)=(x-4)^2$ with domain $x\le 4$.

    $f^{-1}(x)=4-\sqrt{x}$. For $x\le 4$, use the negative square root.

13. What is $f^{-1}(x)$ for $f(x)=(x+1)^2+5$ with restricted domain $x\le -1$?

    $f^{-1}(x)=-1-\sqrt{x-5}$. Solve $y=(x+1)^2+5$ for $x$ using the negative branch.

14. Identify the restricted domain that makes $f(x)=-(x-1)^2+7$ invertible using the right branch.

    Restrict to $x\ge 1$. The vertex is at $x=1$, so restrict to the right side.

15. What test determines whether a function is one-to-one using horizontal lines?

    Horizontal line test: each $y$ hits graph once. If any horizontal line crosses the graph more than once, the function is not one-to-one.

16. What restriction makes $f(x)=x^2-4$ invertible to match the principal square root branch?

    Restrict to $x\ge 0$. The vertex is at $x=0$, so restrict to the right side.

17. What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the left branch?

    Restrict to $x\le h$. The vertex $x=h$ is the axis of symmetry of the parabola.

18. What restriction makes $f(x)=(x-3)^2$ invertible to match the principal square root branch?

    Restrict to $x\ge 3$. The vertex is at $x=3$, so restrict to the right side.

19. What is the inverse of $f(x)=|x|$ when restricted to $x\ge 0$?

    $f^{-1}(x)=x$ for $x\ge 0$. When $x\ge 0$, $f(x) = |x| = x$, so $f^{-1}(x) = x$.

20. What is the range of $f(x)=(x-3)^2$ when the domain is restricted to $x\ge 3$?

    Range is $y\ge 0$. The parabola opens upward with vertex at $(3,0)$.

21. Which domain restriction makes $f(x)=x^2$ invertible and matches the principal square root?

    Restrict to $x\ge 0$. This restriction uses the right branch of the parabola.

22. What restriction makes $f(x)=-(x+2)^2$ invertible using the right half of the parabola?

    Restrict to $x\ge -2$. The vertex is at $x=-2$, so restrict to the right side.

23. What is the range of $f^{-1}(x)$ if $f$ is restricted to the domain $x\le 3$?

    Range of $f^{-1}$ is $x\le 3$. The range of the inverse equals the domain of the original function.

24. What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\ge 1$?

    $f^{-1}(x)=1+\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the right branch.

25. What is the goal of restricting a domain to make a function invertible?

    Make the function one-to-one. Only one-to-one functions have inverses that are also functions.

26. What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\ge -2$?

    $f^{-1}(x)=-2+\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the positive branch.

27. What is the range of $f(x)=x^2$ when the domain is restricted to $x\ge 0$?

    Range is $y\ge 0$. Squaring non-negative inputs produces non-negative outputs.

28. What is $f^{-1}(x)$ for $f(x)=(x+1)^2+5$ with restricted domain $x\ge -1$?

    $f^{-1}(x)=-1+\sqrt{x-5}$. Solve $y=(x+1)^2+5$ for $x$ using the positive branch.

29. Identify the restricted domain that makes $f(x)=(x+5)^2$ invertible using the left branch.

    Restrict to $x\le -5$. The vertex is at $x=-5$, so restrict to the left side.

30. What must you check after solving for $y$ when finding an inverse of a restricted quadratic?

    Choose the correct sign to match the restriction. The sign must correspond to the restricted domain interval.

31. What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\le 3$?

    $f^{-1}(x)=3-\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the negative square root.

32. What is the first algebraic step to find an inverse after restricting the domain?

    Write $y=f(x)$ and swap $x$ and $y$. This sets up the equation to solve for the inverse.

33. What is the inverse of $f(x)=|x|$ when restricted to $x \geq 0$?

    $f^{-1}(x)=x$ for $x \geq 0$. When $x \geq 0$, $f(x) = |x| = x$, so $f^{-1}(x) = x$.

34. What is the inverse of $f(x)=x^2$ when the domain is restricted to $x\le 0$?

    $f^{-1}(x)=-\sqrt{x}$. The negative square root matches the left branch restriction.

35. Which restriction makes $f(x)=x^2+2x$ invertible by using the decreasing side of the parabola?

    Restrict to $x\le -1$. Complete the square to find vertex at $x=-1$, then use left side.

36. What is the inverse of $f(x)=x^2$ if the restricted domain is $[-2,0]$?

    $f^{-1}(x)=-\sqrt{x}$ for $0\le x\le 4$. The range $[0,4]$ comes from squaring the domain $[-2,0]$.

37. What happens to domain and range when taking an inverse $f^{-1}$?

    Domain and range swap. The input and output sets exchange roles.

38. What is the inverse of $f(x)=|x|$ when restricted to $x\le 0$?

    $f^{-1}(x)=-x$ for $x\ge 0$. When $x\le 0$, $f(x) = |x| = -x$, so $f^{-1}(x) = -x$.

39. What restriction makes $f(x)=-(x-h)^2+k$ one-to-one using the right branch?

    Restrict to $x\ge h$. The vertex $x=h$ is the axis of symmetry of the parabola.

40. What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the right branch?

    Restrict to $x\ge h$. The vertex $x=h$ is the axis of symmetry of the parabola.

41. What is $f^{-1}(x)$ for $f(x)=(x+5)^2$ with restricted domain $x\le -5$?

    $f^{-1}(x)=-5-\sqrt{x}$. Solve $y=(x+5)^2$ for $x$ using the negative branch.

42. What does restricting the domain of a function mean?

    Limit inputs to a subset of the original domain. This creates a new function with a smaller domain.

43. What is the domain of $f^{-1}(x)$ if $f$ is restricted to be one-to-one with range $y\ge 5$?

    Domain of $f^{-1}$ is $x\ge 5$. The domain of the inverse equals the range of the original function.

44. Identify the correct inverse for restricted $f(x)=(x-4)^2$ with domain $x\ge 4$.

    $f^{-1}(x)=4+\sqrt{x}$. For $x\ge 4$, use the positive square root.

45. What restriction makes $f(x)=-(x-h)^2+k$ one-to-one using the left branch?

    Restrict to $x\le h$. The vertex $x=h$ is the axis of symmetry of the parabola.

46. What restriction makes $f(x)=(x+1)^2+5$ invertible using the increasing side?

    Restrict to $x\ge -1$. The vertex is at $x=-1$, so restrict to the right side.

47. Which restriction makes $f(x)=x^2+2x$ invertible by using the increasing side of the parabola?

    Restrict to $x\ge -1$. Complete the square to find vertex at $x=-1$, then use right side.

48. Identify the restricted domain that makes $f(x)=x^2$ invertible if the domain is $[-2,2]$.

    Restrict to $[0,2]$ or $[-2,0]$. Choose one monotonic piece from each side of the vertex at $x=0$.

49. What is the relationship between the graphs of $f$ and $f^{-1}$?

    They reflect across the line $y=x$. The graphs are mirror images across the diagonal line $y=x$.

50. What does it mean for a function $f$ to be one-to-one on its domain?

    $f(a)=f(b)\Rightarrow a=b$. Each input maps to exactly one output.

51. What is the inverse of $f(x)=x^2$ if the restricted domain is $[0,2]$?

    $f^{-1}(x)=\sqrt{x}$ for $0\le x\le 4$. The range $[0,4]$ comes from squaring the domain $[0,2]$.

52. What is the inverse of $f(x)=x^2$ when the domain is restricted to $x\ge 0$?

    $f^{-1}(x)=\sqrt{x}$. The positive square root matches the right branch restriction.

Study these flashcards