Algebra 2 Flashcards: Verify Functions Are Inverses

What this deck covers

This deck focuses on Verify Functions Are Inverses, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 38: Which statement correctly describes inverses using composition notation?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 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 $\text{Dom}(f)$.

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

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

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

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

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

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

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

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

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

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

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