Verify Functions Are Inverses - Algebra 2

1. What is $g(f(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?

   $g(f(x))=x$. Substituting: $g(x^3+1)=\sqrt[3]{(x^3+1)-1}=x$.

2. What is the key simplification goal when checking $f(g(x))$ to verify inverses?

   Simplify until the result is exactly $x$. The composition should reduce to the identity.

3. What is $f(g(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $g$?

   $f(g(x))=x$ for $x\ge 0$. Substituting: $f(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.

4. What is $f(g(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?

   $f(g(x))=x$ for $x>0$. Domain restriction needed for $\log_2(x)$.

5. What is $g(f(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $f$?

   $g(f(x))=x$ for $x\ge 0$. Substituting: $g(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.

6. Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $g(f(x))$.

   $g(f(x))=x$. Substituting: $g(x+7)=(x+7)-7=x$.

7. What is $f(g(x))$ if $f(x)=x-\frac{3}{x}$ and $g(x)=x+\frac{3}{x}$ for $x\ne 0$?

   $f(g(x))\ne x$ (so they are not inverses). These functions don't compose to the identity.

8. What is $g(f(x))$ if $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$?

   $g(f(x))=x$. Substituting: $g(x^3)=\sqrt[3]{x^3}=x$.

9. What is $g(f(x))$ if $f(x)=\frac{4}{x}$ and $g(x)=\frac{4}{x}$ with $x\ne 0$?

   $g(f(x))=x$ for $x\ne 0$. Substituting: $g(\frac{4}{x})=\frac{4}{\frac{4}{x}}=x$.

10. Identify $g(f(x))$ if $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x}$ with $x\ne 0$.

    $g(f(x))=x$ for $x\ne 0$. Substituting: $g(\frac{1}{x})=\frac{1}{\frac{1}{x}}=x$.

11. What is the identity function used in inverse verification by composition?

    $I(x)=x$. The function that returns its input unchanged.

12. Identify $f(g(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.

    $f(g(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.

13. What is $g(f(x))$ if $f(x)=x^5$ and $g(x)=\sqrt[5]{x}$?

    $g(f(x))=x$. Substituting: $g(x^5)=\sqrt[5]{x^5}=x$.

14. What is $g(f(x))$ if $f(x)=\log_5(x)$ and $g(x)=5^x$?

    $g(f(x))=x$ for $x>0$. Domain restriction needed for $\log_5(x)$.

15. What is $g(f(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?

    $g(f(x))=x$. Exponential and logarithm base 2 are inverses.

16. What is $f(g(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?

    $f(g(x))=x$. Substituting: $f(\frac{x+5}{2})=2\cdot\frac{x+5}{2}-5=x$.

17. What is the inverse relationship between domain and range for inverse functions?

    $\text{Dom}(f)=\text{Ran}(g)$ and $\text{Ran}(f)=\text{Dom}(g)$. Domain and range swap between inverse functions.

18. What does it mean if $f(g(x))=x$ but $g(f(x))\neq x$ for some $x$ in the domain?

    $f$ and $g$ are not inverses (on those domains). Both compositions must equal $x$ to be true inverses.

19. What must be true about $f(g(x))$ and $g(f(x))$ to verify $g$ is the inverse of $f$?

    $f(g(x))=x$ and $g(f(x))=x$ (on the appropriate domains). Both compositions must equal the identity function.

20. What is $f(g(x))$ if $f(x)=\log_5(x)$ and $g(x)=5^x$?

    $f(g(x))=x$. Logarithm and exponential base 5 are inverses.

21. What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?

    $f(g(x))=x$ for $x\ge 0$. Substituting: $f(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.

22. What is $g(f(x))$ if $f(x)=\frac{x-4}{5}$ and $g(x)=5x+4$?

    $g(f(x))=x$. Substituting: $g(\frac{x-4}{5})=5\cdot\frac{x-4}{5}+4=x$.

23. Identify $f(g(x))$ if $f(x)=\ln(x)$ and $g(x)=e^x$.

    $f(g(x))=x$. Natural log and exponential are inverses.

24. What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?

    $g(f(x))=x$ for $x\ge 0$. Substituting: $g(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.

25. Identify the conclusion if $f(g(x))=x$ but only for $x>0$, and $g(f(x))=x$ for all real $x$.

    They are inverses only with domain restriction $x>0$ where needed. Domain restrictions must match for inverse relationship.

26. What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ without restricting $f$ to $x\ge 0$?

    $f(g(x))=x$ for $x\ge 0$ only. Without domain restriction, $f$ isn't one-to-one.

27. What must be checked about domains when verifying inverses by composition?

    Each composition must equal $x$ for all $x$ in its stated domain. Compositions must work on their proper domains.

28. What test is commonly used to decide whether a function is one-to-one before finding an inverse?

    The horizontal line test. Checks if every horizontal line intersects once.

29. Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $f(g(x))$.

    $f(g(x))=x$. Substituting: $f(x-7)=(x-7)+7=x$.

30. What is $f(g(x))$ if $f(x)=x^5$ and $g(x)=\sqrt[5]{x}$?

    $f(g(x))=x$. Substituting: $f(\sqrt[5]{x})=(\sqrt[5]{x})^5=x$.

31. Identify $g(f(x))$ if $f(x)=\ln(x)$ and $g(x)=e^x$.

    $g(f(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.

32. What is $g(f(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?

    $g(f(x))=x$. Substituting: $g(3x)=\frac{3x}{3}=x$.

33. Identify $g(f(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.

    $g(f(x))=x$. Exponential and natural log are inverses.

34. What is $g(f(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?

    $g(f(x))=x$. Substituting: $g(2x-5)=\frac{(2x-5)+5}{2}=x$.

35. What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ without restricting $f$ to $x\ge 0$?

    $g(f(x))=|x|$. Square root returns absolute value for all real inputs.

36. What is $f(g(x))$ if $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$?

    $f(g(x))=x$. Substituting: $f(\sqrt[3]{x})=(\sqrt[3]{x})^3=x$.

37. What is a common reason $f$ has no inverse function without restrictions?

    $f$ is not one-to-one on its domain. Without one-to-one property, no inverse exists.

38. Which statement correctly describes inverses using composition notation?

    $f\circ g=I$ and $g\circ f=I$. Composition notation for inverse relationships.

39. What is $f(g(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?

    $f(g(x))=x$. Substituting: $f(\frac{x}{3})=3\cdot\frac{x}{3}=x$.

40. What is $f(g(x))$ if $f(x)=\frac{x-4}{5}$ and $g(x)=5x+4$?

    $f(g(x))=x$. Substituting: $f(5x+4)=\frac{(5x+4)-4}{5}=x$.

41. What is the typical algebraic first step before composing to verify inverses for formulas?

    Substitute one function into the other: compute $f(g(x))$ and $g(f(x))$. Replace variable in one function with the other.

42. What coordinate swap describes the inverse relationship between points on $f$ and $f^{-1}$?

    If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$. Coordinates swap between function and inverse.

43. What single composition is sufficient to verify inverses when domains and ranges match appropriately?

    Either $f(g(x))=x$ or $g(f(x))=x$ (with correct domain restrictions). One composition suffices when domains match ranges.

44. What is $g(f(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?

    $g(f(x))=x$. Substituting: $g(\frac{x-1}{2})=2\cdot\frac{x-1}{2}+1=x$.

45. What is $f(g(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?

    $f(g(x))=x$. Substituting: $f(\sqrt[3]{x-1})=(\sqrt[3]{x-1})^3+1=x$.

46. Identify the conclusion if $f(g(x))=x$ for all $x$ in $\text{Dom}(g)$ and $g(f(x))=x$ for all $x$ in $\text{Dom}(f)$.

    $g=f^{-1}$ and $f=g^{-1}$ (with those domains). Both compositions equal identity on proper domains.

47. What is the conclusion if simplifying $f(g(x))$ produces $x$ only when $x\ge 2$?

    They are inverses only on the restricted domain $x\ge 2$. Inverses only exist on the restricted domain.

48. What is $f(g(x))$ if $f(x)=\frac{4}{x}$ and $g(x)=\frac{4}{x}$ with $x\ne 0$?

    $f(g(x))=x$ for $x\ne 0$. Substituting: $f(\frac{4}{x})=\frac{4}{\frac{4}{x}}=x$.

49. Identify $f(g(x))$ if $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x}$ with $x\ne 0$.

    $f(g(x))=x$ for $x\ne 0$. Substituting: $f(\frac{1}{x})=\frac{1}{\frac{1}{x}}=x$.

50. What is $f(g(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?

    $f(g(x))=x$. Substituting: $f(2x+1)=\frac{(2x+1)-1}{2}=x$.

51. What must be true about the graph of $f$ and the graph of $f^{-1}$?

    They are reflections across the line $y=x$. Inverse functions reflect across $y=x$.

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