AP Precalculus Flashcards: Inverse Functions

What this deck covers

This deck focuses on Inverse Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: $f$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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