Read Inverses from Graphs or Tables - Algebra 2

1. Which value equals $f^{-1}(y)$ if a table shows the pair $(x,y)=(7,20)$ for $f$?

   $f^{-1}(20)=7$. The inverse reverses the input-output pair $(7,20)$.

2. What is $f^{-1}(3)$ if the graph of $f$ includes $( -1,3)$?

   $f^{-1}(3)=-1$. Read the $x$-coordinate where the graph has $y=3$.

3. Identify the point on $f^{-1}$ that corresponds to the $y$-intercept $(0,c)$ of $f$.

   The point $(c,0)$ on $f^{-1}$. The $y$-intercept of $f$ becomes the $x$-intercept of $f^{-1}$.

4. Identify $f^{-1}(0)$ if a graph of $f$ crosses the $y$-axis at $(0,7)$.

   $f^{-1}(0)$ is not determined from $(0,7)$ alone. The $y$-intercept alone doesn't show where $f(x)=0$.

5. What is the $y$-intercept of $f^{-1}$ if the $x$-intercept of $f$ is $(c,0)$?

   The $y$-intercept of $f^{-1}$ is $(0,c)$. Intercepts swap coordinates when finding the inverse.

6. What property must $f$ have (on its domain) to have an inverse function?

   $f$ must be one-to-one. Each output must correspond to exactly one input.

7. What is the domain of $f^{-1}$ in terms of $f$?

   The domain of $f^{-1}$ is the range of $f$. The inverse's inputs are the original function's outputs.

8. What is $f^{-1}(\frac{3}{2})$ if the graph of $f$ includes $(4,\frac{3}{2})$?

   $f^{-1}(\frac{3}{2})=4$. Read the $x$-coordinate where the graph has $y=\frac{3}{2}$.

9. What is $f^{-1}(-\frac{3}{4})$ if a table shows $f(2)=-\frac{3}{4}$?

   $f^{-1}(-\frac{3}{4})=2$. The inverse reverses the input-output pair $(2,-\frac{3}{4})$.

10. What is $f^{-1}(9)$ if a graph of $f$ shows the point $(1,9)$?

    $f^{-1}(9)=1$. Read the $x$-coordinate where the graph has $y=9$.

11. What is $f^{-1}(6)$ if a graph of $f$ contains the point $(4,6)$?

    $f^{-1}(6)=4$. Read the $x$-coordinate where the graph has $y=6$.

12. What point lies on $f^{-1}$ if the point $(-5,8)$ lies on $f$?

    $(8,-5)$ lies on $f^{-1}$. Inverse functions swap the coordinates of points.

13. Identify the range of $f$ if the domain of $f^{-1}$ is $[-2,5]$.

    The range of $f$ is $[-2,5]$. Domain and range swap between inverse functions.

14. What is $f^{-1}(1)$ if the graph of $f$ includes $(\frac{1}{2},1)$?

    $f^{-1}(1)=\frac{1}{2}$. Read the $x$-coordinate where the graph has $y=1$.

15. What is $f^{-1}(a)$ if $f(4)=a$?

    $f^{-1}(a)=4$. The inverse reverses any input-output relationship.

16. What happens to the domain and range when passing from $f$ to $f^{-1}$?

    Domain and range swap between $f$ and $f^{-1}$. The input set of one becomes the output set of the other.

17. What is $f^{-1}(5)$ if a graph of $f$ has the point $(-6,5)$?

    $f^{-1}(5)=-6$. Read the $x$-coordinate where the graph has $y=5$.

18. What equation states the composition identity for inverses using $f(f^{-1}(x))$?

    $f(f^{-1}(x))=x$ (for $x$ in the domain of $f^{-1}$). Composing a function with its inverse yields the identity.

19. Which ordered pair on $f$ corresponds to the point $(2,9)$ on $f^{-1}$?

    $(9,2)$ on $f$. Inverse functions swap coordinates of corresponding points.

20. Identify the meaning of the statement $f^{-1}(y)=x$ in terms of $f$.

    It means $f(x)=y$. The inverse notation shows which input produces output $y$.

21. What is the value of $f^{-1}(b)$ if $f(a)=b$?

    $f^{-1}(b)=a$. Inverse functions reverse input-output relationships.

22. Identify the point on $f^{-1}$ that corresponds to the $x$-intercept $(c,0)$ of $f$.

    The point $(0,c)$ on $f^{-1}$. The $x$-intercept of $f$ becomes the $y$-intercept of $f^{-1}$.

23. What is $f^{-1}(x)$ at $x=-8$ if the table shows $f(0)=-8$?

    $f^{-1}(-8)=0$. The inverse reverses the input-output pair $(0,-8)$.

24. What point lies on $f^{-1}$ if the point $(0,-7)$ lies on $f$?

    $(-7,0)$ lies on $f^{-1}$. Inverse functions swap the coordinates of points.

25. What is $f^{-1}(12)$ if a table shows $f(9)=12$?

    $f^{-1}(12)=9$. The inverse reverses the input-output pair $(9,12)$.

26. Which ordered pair on $f^{-1}$ corresponds to the point $(-3,11)$ on $f$?

    $(11,-3)$ on $f^{-1}$. Inverse functions swap coordinates of corresponding points.

27. What is the value of $f(a)$ if $f^{-1}(b)=a$?

    $f(a)=b$. If the inverse maps $b$ to $a$, then $f$ maps $a$ to $b$.

28. What is $f^{-1}(0)$ if a table shows $f(-4)=0$?

    $f^{-1}(0)=-4$. The inverse reverses the input-output pair $(-4,0)$.

29. What is $f^{-1}(\frac{1}{2})$ if a table shows $f(8)=\frac{1}{2}$?

    $f^{-1}(\frac{1}{2})=8$. The inverse reverses the input-output pair $(8,\frac{1}{2})$.

30. What is the range of $f^{-1}$ in terms of $f$?

    The range of $f^{-1}$ is the domain of $f$. The inverse's outputs are the original function's inputs.

31. What is $f^{-1}(-6)$ if the graph of $f$ includes $(2,-6)$?

    $f^{-1}(-6)=2$. Read the $x$-coordinate where the graph has $y=-6$.

32. What equation states the composition identity for inverses using $f^{-1}(f(x))$?

    $f^{-1}(f(x))=x$ (for $x$ in the domain of $f$). Composing an inverse with its function yields the identity.

33. What is $f^{-1}(-2)$ if the graph of $f$ includes $(9,-2)$?

    $f^{-1}(-2)=9$. Read the $x$-coordinate where the graph has $y=-2$.

34. What is the $x$-intercept of $f^{-1}$ if the $y$-intercept of $f$ is $(0,c)$?

    The $x$-intercept of $f^{-1}$ is $(c,0)$. Intercepts swap coordinates when finding the inverse.

35. What is the key coordinate relationship between $f$ and $f^{-1}$ on a graph?

    If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$. Inverse functions swap $x$ and $y$ coordinates.

36. What is $f^{-1}(-3)$ if a table shows $f(5)=-3$?

    $f^{-1}(-3)=5$. The inverse reverses the input-output pair $(5,-3)$.

37. What is $f^{-1}(1)$ if a table shows $f(0)=1$?

    $f^{-1}(1)=0$. The inverse reverses the input-output pair $(0,1)$.

38. What point lies on $f^{-1}$ if the point $(3,-2)$ lies on $f$?

    $( -2,3)$ lies on $f^{-1}$. Inverse functions swap the coordinates of points.

39. What graph test checks whether a function is one-to-one?

    The horizontal line test. No horizontal line intersects the graph more than once.

40. What is $f^{-1}(b)$ if $f(-9)=b$?

    $f^{-1}(b)=-9$. The inverse reverses any input-output relationship.

41. What is $f^{-1}(x)$ at $x=4$ if the table shows $f(1)=4$?

    $f^{-1}(4)=1$. The inverse reverses the input-output pair $(1,4)$.

42. Identify $f^{-1}(0)$ if a graph of $f$ crosses the $x$-axis at $(7,0)$.

    $f^{-1}(0)=7$. The $x$-intercept shows where $f(x)=0$.

43. What is $f^{-1}(2)$ if a table shows $f(-1)=2$?

    $f^{-1}(2)=-1$. The inverse reverses the input-output pair $(-1,2)$.

44. What is $f^{-1}(10)$ if a table shows $f(3)=10$?

    $f^{-1}(10)=3$. The inverse reverses the input-output pair $(3,10)$.

45. What is $f^{-1}(13)$ if the graph of $f$ includes $(0,13)$?

    $f^{-1}(13)=0$. Read the $x$-coordinate where the graph has $y=13$.

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

    The line $y=x$. Functions and their inverses are reflections across this diagonal.

47. What is the value of $f^{-1}(b)$ if the graph shows the point $(b,a)$ on $f^{-1}$?

    $f^{-1}(b)=a$. Reading coordinates directly from the inverse graph.

48. What is $f^{-1}(7)$ if a table shows $f(2)=7$?

    $f^{-1}(7)=2$. The inverse reverses the input-output pair $(2,7)$.

49. What is $f^{-1}(4)$ if a graph of $f$ shows $f(-2)=4$?

    $f^{-1}(4)=-2$. Find the input that produces output $4$.

50. What is $f^{-1}(-1)$ if a graph of $f$ shows $f(6)=-1$?

    $f^{-1}(-1)=6$. Find the input that produces output $-1$.

51. Which value equals $f^{-1}(y)$ if a table shows the pair $(x,y)=(-2,-9)$ for $f$?

    $f^{-1}(-9)=-2$. The inverse reverses the input-output pair $(-2,-9)$.

52. What is the value of $f(a)$ if the graph shows the point $(a,b)$ on $f$?

    $f(a)=b$. Reading coordinates directly from the function graph.

53. Identify the domain of $f$ if the range of $f^{-1}$ is $(0,10)$.

    The domain of $f$ is $(0,10)$. Domain and range swap between inverse functions.

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