Find and Write an Inverse Function - Algebra 2

1. What domain restriction makes $f(x)=x^2$ invertible as a function?

   Restrict to $x\ge 0$ or restrict to $x\le 0$. These restrictions make $f(x)=x^2$ one-to-one.

2. What is the inverse of $f(x)=\frac{x+4}{2}$?

   $f^{-1}(x)=2x-4$. Multiply by 2, then subtract 4.

3. What identity must be true if $f$ and $f^{-1}$ are inverses?

   $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$. These compositions verify the inverse relationship.

4. What is the domain restriction for $f(x)=\frac{x+1}{x-1}$?

   $x\neq 1$. Domain excludes values making denominator zero.

5. What is the inverse of $f(x)=\frac{x-6}{2}$?

   $f^{-1}(x)=2x+6$. Multiply by 2, then add 6.

6. What is the solution for $x$ in $f(x)=c$ if $f(x)=ax+b$ and $a\neq 0$?

   $x=\frac{c-b}{a}$. Subtract $b$, then divide by $a$.

7. What is the inverse of $f(x)=x^3$?

   $f^{-1}(x)=\sqrt[3]{x}$. Cubing and cube root are inverse operations.

8. What restriction on $c$ is needed to solve $\frac{x+1}{x-1}=c$ for $x$?

   $c\neq 1$. Denominator cannot equal zero in rational function.

9. What is the inverse of $f(x)=\frac{x+1}{x-1}$ with $x\neq 1$?

   $f^{-1}(x)=\frac{x+1}{x-1}$ with $x\neq 1$. This rational function is its own inverse.

10. What is the solution for $x$ in $f(x)=c$ if $f(x)=\frac{x+1}{x-1}$ and $c\neq 1$?

    $x=\frac{c+1}{c-1}$. Cross-multiply and solve for $x$.

11. What is the inverse of $f(x)=\frac{5-x}{3}$?

    $f^{-1}(x)=5-3x$. Subtract from 5, then divide by 3.

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

    $x\neq 1$. Domain of inverse equals range of original function.

13. What is the inverse of $f(x)=(x-3)^3$?

    $f^{-1}(x)=\sqrt[3]{x}+3$. Take cube root, then add 3.

14. What is the coordinate relationship between inverse functions on a graph?

    Points $(a,b)$ on $f$ become $(b,a)$ on $f^{-1}$. Coordinates are reflected across $y=x$.

15. What test is commonly used to check if a graph represents a one-to-one function?

    The horizontal line test. Each horizontal line intersects the graph at most once.

16. What is the inverse of $f(x)=\frac{x}{x+1}$ with $x\neq -1$?

    $f^{-1}(x)=\frac{x}{1-x}$ with $x\neq 1$. Cross-multiply and solve for $y$ after swapping variables.

17. What is the inverse of the function $f(x)=x-a$?

    $f^{-1}(x)=x+a$. Addition and subtraction are inverse operations.

18. What is the inverse of $f(x)=\frac{3}{x}$ with $x\neq 0$?

    $f^{-1}(x)=\frac{3}{x}$ with $x\neq 0$. Reciprocal function with constant numerator.

19. What is the inverse of $f(x)=\sqrt[3]{4x}$?

    $f^{-1}(x)=\frac{x^3}{4}$. Cube the input, then divide by 4.

20. What is the inverse of $f(x)=2x^3$?

    $f^{-1}(x)=\sqrt[3]{\frac{x}{2}}$. Divide by 2, then take cube root.

21. What is the inverse of $f(x)=3x-7$?

    $f^{-1}(x)=\frac{x+7}{3}$. Add 7, then divide by 3.

22. What is the inverse of $f(x)=x^3-1$?

    $f^{-1}(x)=\sqrt[3]{x+1}$. Add 1, then take cube root.

23. What is the key step to start finding $f^{-1}(x)$ from $y=f(x)$?

    Swap $x$ and $y$ in $y=f(x)$. This is the first step in the standard algebraic method.

24. What is the inverse of $f(x)=\frac{x-1}{2x+3}$ with $x\neq -\frac{3}{2}$?

    $f^{-1}(x)=\frac{1+3x}{1-2x}$ with $x\neq \frac{1}{2}$. Cross-multiply and solve for $y$ after swapping variables.

25. What is the inverse of the function $f(x)=ax$ where $a\neq 0$?

    $f^{-1}(x)=\frac{x}{a}$. Multiplication and division are inverse operations.

26. What is $f^{-1}(x)$ if $f(x)=\frac{x-7}{x+2}$ with $x\neq -2$?

    $f^{-1}(x)=\frac{2x+7}{1-x}$ with $x\neq 1$. Cross-multiply and solve for $y$ after swapping variables.

27. What is the inverse of $f(x)=ax+b$ where $a\neq 0$?

    $f^{-1}(x)=\frac{x-b}{a}$. Apply inverse operations in reverse order.

28. What is the solution for $x$ in $f(x)=c$ if $f(x)=x^3-4$?

    $x=\sqrt[3]{c+4}$. Add 4, then take cube root.

29. What must you do after swapping $x$ and $y$ to find $f^{-1}(x)$?

    Solve the new equation for $y$. This isolates $y$ to express the inverse function.

30. What is the range of $f^{-1}(x)$ if $f(x)=\frac{x+1}{x-1}$?

    $y\neq 1$. Range of inverse equals domain of original function.

31. What is the inverse of $f(x)=\sqrt[n]{x}$ for odd $n$?

    $f^{-1}(x)=x^n$. Odd roots and powers are inverse operations.

32. What is the inverse of $f(x)=-2x^3+6$?

    $f^{-1}(x)=\sqrt[3]{\frac{6-x}{2}}$. Subtract from 6, divide by 2, then take cube root.

33. What is the inverse of the function $f(x)=x+a$?

    $f^{-1}(x)=x-a$. Addition and subtraction are inverse operations.

34. What is the inverse of $f(x)=\frac{2x+3}{x-4}$ with $x\neq 4$?

    $f^{-1}(x)=\frac{4x+3}{x-2}$ with $x\neq 2$. Cross-multiply and solve for $y$ after swapping variables.

35. What is the inverse of $f(x)=x^5$?

    $f^{-1}(x)=\sqrt[5]{x}$. Fifth power and fifth root are inverse operations.

36. What is the inverse of $f(x)=\sqrt[3]{x+8}$?

    $f^{-1}(x)=x^3-8$. Cube the input, then subtract 8.

37. What is the inverse of $f(x)=\frac{2x}{3}+1$?

    $f^{-1}(x)=\frac{3}{2}(x-1)$. Subtract 1, then multiply by $\frac{3}{2}$.

38. What is $f^{-1}(x)$ if $f(x)=\frac{x+3}{x-2}$ with $x\neq 2$?

    $f^{-1}(x)=\frac{2x+3}{x-1}$ with $x\neq 1$. Cross-multiply and solve for $y$ after swapping variables.

39. What is the inverse of $f(x)=7-4x$?

    $f^{-1}(x)=\frac{7-x}{4}$. Subtract from 7, then divide by 4.

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

    $f^{-1}(x)=-\sqrt{x}$. Square and negative square root are inverses on non-positive domain.

41. What is the range restriction for $f(x)=\frac{x+1}{x-1}$?

    $y\neq 1$. Range excludes horizontal asymptote value.

42. What is the inverse of $f(x)=-5x+10$?

    $f^{-1}(x)=\frac{10-x}{5}$. Subtract from 10, then divide by 5.

43. What condition must $f$ satisfy to have an inverse function on its domain?

    $f$ must be one-to-one on its domain. Only one-to-one functions pass the horizontal line test.

44. What is the inverse of $f(x)=\frac{x+1}{2}$?

    $f^{-1}(x)=2x-1$. Multiply by 2, then subtract 1.

45. What is the inverse of $f(x)=x^3+4$?

    $f^{-1}(x)=\sqrt[3]{x-4}$. Apply cube root, then subtract 4.

46. What is the inverse of $f(x)=\sqrt[3]{x}$?

    $f^{-1}(x)=x^3$. Cube root and cubing are inverse operations.

47. What is the inverse of $f(x)=\frac{4x-1}{2x+5}$ with $x\neq -\frac{5}{2}$?

    $f^{-1}(x)=\frac{1+5x}{4-2x}$ with $x\neq 2$. Cross-multiply and solve for $y$ after swapping variables.

48. What line reflects the graph of $y=f(x)$ to get the graph of $y=f^{-1}(x)$?

    The line $y=x$. Graphs of inverse functions are reflections across this line.

49. What is the inverse of $f(x)=(x+2)^3-5$?

    $f^{-1}(x)=\sqrt[3]{x+5}-2$. Add 5, take cube root, then subtract 2.

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

    $f^{-1}(x)=\sqrt{x}$. Square and square root are inverses on non-negative domain.

51. What is the inverse of $f(x)=\frac{x-2}{x+5}$ with $x\neq -5$?

    $f^{-1}(x)=\frac{5x+2}{1-x}$ with $x\neq 1$. Cross-multiply and solve for $y$ after swapping variables.

52. What is the solution for $x$ in $f(x)=c$ if $f(x)=2x^3$?

    $x=\sqrt[3]{\frac{c}{2}}$. Divide $c$ by 2, then take cube root.

53. What is the inverse of $f(x)=\frac{1}{x}$ with domain $x\neq 0$?

    $f^{-1}(x)=\frac{1}{x}$ with $x\neq 0$. Reciprocal function is its own inverse.

54. What is the inverse of $f(x)=\frac{x-4}{3}$?

    $f^{-1}(x)=3x+4$. Multiply by 3, then add 4.

55. What is the inverse of $f(x)=\sqrt{x}$ with domain $x\ge 0$?

    $f^{-1}(x)=x^2$ with domain $x\ge 0$. Square root and square are inverse operations.

56. What is the solution for $x$ in $f(x)=c$ if $f(x)=\frac{1}{x}$ and $c\neq 0$?

    $x=\frac{1}{c}$. Take reciprocal of $c$.

57. What is the relationship between the domains and ranges of $f$ and $f^{-1}$?

    Domain of $f^{-1}$ = range of $f$, and vice versa. Input and output sets are swapped between inverse functions.

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