Inverse Functions - AP Precalculus

1. What is the inverse of the logarithmic function $f(x)=\log_a(x)$ for $a>0$ and $a\ne 1$?

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

2. What is the standard algebraic procedure to find $f^{-1}(x)$ from $y=f(x)$?

   Swap $x$ and $y$, then solve for $y$ and rename $y$ as $f^{-1}(x)$. This process reverses the input-output relationship of $f$.

3. Identify the inverse of $f(x)=\ln(x)$.

   $f^{-1}(x)=e^x$. Natural exponential is the inverse of natural log.

4. Identify the inverse of $f(x)=e^x$.

   $f^{-1}(x)=\ln(x)$. Natural log is the inverse of the natural exponential.

5. Identify a domain restriction that makes $f(x)=x^2$ have an inverse function.

   Restrict to $[0,\infty)$ or to $(-\infty,0]$. On these intervals, $f(x)=x^2$ is strictly monotonic.

6. Identify the inverse of $f(x)=(x-2)^3+1$.

   $f^{-1}(x)=\sqrt[3]{x-1}+2$. Undo operations in reverse order: subtract 1, take cube root, add 2.

7. Find $f^{-1}(x)$ for $f(x)=3x-5$.

   $f^{-1}(x)=\frac{x+5}{3}$. Swap: $x=3y-5$, solve: $y=\frac{x+5}{3}$.

8. Identify the inverse function of $f(x)=3x-5$.

   $f^{-1}(x)=\frac{x+5}{3}$. Swap $x$ and $y$, then solve for $y$.

9. What graphical transformation relates the graphs of $y=f(x)$ and $y=f^{-1}(x)$?

   Reflection across the line $y=x$. Points $(a,b)$ and $(b,a)$ are reflections across this line.

10. What is the inverse of the point $(a,b)$ on $y=f(x)$ as a point on $y=f^{-1}(x)$?

    $(b,a)$. Swapping coordinates reflects the point across $y=x$.

11. What is the definition of an inverse function in terms of composition?

    $f^{-1}(f(x))=x$ and $f(f^{-1}(x))=x$ on the appropriate domains. These compositions yield the identity function on their respective domains.

12. What is the correct meaning of the notation $f^{-1}(x)$?

    It is the inverse function of $f$, not the reciprocal $\frac{1}{f(x)}$. The $-1$ is an exponent notation, not a negative power.

13. What condition must a function satisfy to have an inverse function that is also a function?

    It must be one-to-one (pass the horizontal line test). Each output must correspond to exactly one input.

14. What is the Horizontal Line Test used to determine about a function $f$?

    Whether $f$ is one-to-one (and therefore invertible on its domain). If any horizontal line hits the graph twice, $f$ has no inverse.

15. Identify whether $f(x)=x^2$ is invertible on $(-\infty,\infty)$.

    Not invertible on $(-\infty,\infty)$ (not one-to-one). Both $x=2$ and $x=-2$ give $f(x)=4$, failing one-to-one.

16. What does it mean for a function $f$ to have an inverse function $f^{-1}$?

    $f$ is one-to-one; each $y$ in the range comes from exactly one $x$. One-to-one means no horizontal line intersects the graph more than once.

17. Which restriction makes $f(x)=x^2$ invertible as a function?

    Restrict the domain to $x\ge 0$ (or to $x\le 0$). This makes the function pass the horizontal line test.

18. What is the inverse of the exponential function $f(x)=a^x$ for $a>0$ and $a\ne 1$?

    $f^{-1}(x)=\log_a(x)$. Logarithms and exponentials are inverse operations.

19. Evaluate $f^{-1}(7)$ for $f(x)=2x+1$.

    $f^{-1}(7)=3$. Since $f(3)=7$, we have $f^{-1}(7)=3$.

20. Identify the inverse relation of the equation $y=\frac{2x+3}{5}$.

    $y=\frac{5x-3}{2}$. Swap $x$ and $y$: $x=\frac{2y+3}{5}$, solve for $y$.

21. Find and correct the error: claiming $f^{-1}(x)=\frac{1}{f(x)}$ for an inverse function.

    Correct: $f^{-1}$ is not a reciprocal; it satisfies $f(f^{-1}(x))=x$. The inverse function undoes $f$, not reciprocates it.

22. Given $f(2)=9$, what is $f^{-1}(9)$?

    $f^{-1}(9)=2$. By definition, $f^{-1}$ undoes $f$: if $f(a)=b$, then $f^{-1}(b)=a$.

23. What is the inverse of the linear function $f(x)=mx+b$ with $m\ne 0$?

    $f^{-1}(x)=\frac{x-b}{m}$. Solve $y=mx+b$ for $x$ to get the inverse formula.

24. What happens to a point $(a,b)$ on $y=f(x)$ when graphed on $y=f^{-1}(x)$?

    It becomes the point $(b,a)$. Swapping coordinates reflects the point across $y=x$.

25. Identify the inverse function of $f(x)=\frac{x-4}{7}$.

    $f^{-1}(x)=7x+4$. Multiply by 7 and add 4 to undo the original operations.

26. What is the inverse of the exponential function $f(x)=b^x$ for $b>0$ and $b\ne 1$?

    $f^{-1}(x)=\log_b(x)$. Logarithms and exponentials are inverse operations.

27. What is the inverse of the linear function $f(x)=mx+b$ with $m\ne 0$?

    $f^{-1}(x)=\frac{x-b}{m}$. Solve $y=mx+b$ for $x$ to get $x=\frac{y-b}{m}$.

28. What is the relationship between the domain and range of $f$ and $f^{-1}$?

    $\text{Dom}(f^{-1})=\text{Range}(f)$ and $\text{Range}(f^{-1})=\text{Dom}(f)$. The inverse swaps the input and output sets of $f$.

29. What is the key graph relationship between $y=f(x)$ and $y=f^{-1}(x)$?

    They are reflections across the line $y=x$. Points $(a,b)$ and $(b,a)$ are symmetric about $y=x$.

30. Identify whether $f(x)=x^2$ is one-to-one on the domain $(-\infty,\infty)$.

    No, it is not one-to-one on $(-\infty,\infty)$. It fails the horizontal line test (e.g., $f(-2)=f(2)=4$).

31. Find $f^{-1}(x)$ for $f(x)=\frac{x-4}{2}$.

    $f^{-1}(x)=2x+4$. Swap: $x=\frac{y-4}{2}$, solve: $y=2x+4$.

32. Find $f^{-1}(x)$ for $f(x)=\sqrt{x-1}$ with domain $x\ge 1$.

    $f^{-1}(x)=x^2+1$ with domain $x\ge 0$. Swap: $x=\sqrt{y-1}$, square both sides: $y=x^2+1$.

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

    $f^{-1}(x)=x^2$ with domain $x\ge 0$. Squaring undoes the square root for non-negative values.

34. What is the inverse of the identity function $f(x)=x$?

    $f^{-1}(x)=x$. The identity function is its own inverse since $f(f(x))=x$.

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

    $f^{-1}(x)=\sqrt[3]{x}$. The cube root undoes the cubing operation.

36. What is the composition value $f(f^{-1}(x))$ for inputs $x$ in the domain of $f^{-1}$?

    $f(f^{-1}(x))=x$. By definition of inverse functions.

37. What is the relationship between the domain and range of $f$ and $f^{-1}$?

    $\text{Dom}(f^{-1})=\text{Ran}(f)$ and $\text{Ran}(f^{-1})=\text{Dom}(f)$. The domain and range swap when finding the inverse.

38. What is the defining composition property of inverse functions $f$ and $f^{-1}$?

    $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$ (on appropriate domains). These compositions yield the identity function on their respective domains.

39. Find $f^{-1}(x)$ for $f(x)=(x+2)^3$.

    $f^{-1}(x)=\sqrt[3]{x}-2$. Swap: $x=(y+2)^3$, take cube root: $y=\sqrt[3]{x}-2$.

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