Sign up

Algebra 2 Flashcards: Find And Write An Inverse Function

Study Find And Write An Inverse Function in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra 2

Find And Write An Inverse Function

0 mastered0 still learning

0% Complete

QUESTION
1/ 57

What domain restriction makes f(x)=x2f(x)=x^2 invertible as a function?

Tap card or press Space to flip

ANSWER

Restrict to x≥0x\ge 0 or restrict to x≤0x\le 0. These restrictions make f(x)=x2f(x)=x^2 one-to-one.

How well did you know it?

Card 1 / 57

What this deck covers

This deck focuses on Find And Write An Inverse Function, 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.

All flashcards

Flashcard 1: What domain restriction makes f(x)=x2f(x)=x^2 invertible as a function?

Answer: Restrict to x≥0x\ge 0 or restrict to x≤0x\le 0. These restrictions make f(x)=x2f(x)=x^2 one-to-one.

Flashcard 2: What is the inverse of f(x)=x+42f(x)=\frac{x+4}{2}?

Answer: f−1(x)=2x−4f^{-1}(x)=2x-4. Multiply by 2, then subtract 4.

Flashcard 3: What identity must be true if ff and f−1f^{-1} are inverses?

Answer: f(f−1(x))=xf(f^{-1}(x))=x and f−1(f(x))=xf^{-1}(f(x))=x. These compositions verify the inverse relationship.

Flashcard 4: What is the domain restriction for f(x)=x+1x−1f(x)=\frac{x+1}{x-1}?

Answer: x≠1x\neq 1. Domain excludes values making denominator zero.

Flashcard 5: What is the inverse of f(x)=x−62f(x)=\frac{x-6}{2}?

Answer: f−1(x)=2x+6f^{-1}(x)=2x+6. Multiply by 2, then add 6.

Flashcard 6: What is the solution for xx in f(x)=cf(x)=c if f(x)=ax+bf(x)=ax+b and a≠0a\neq 0?

Answer: x=c−bax=\frac{c-b}{a}. Subtract bb, then divide by aa.

Flashcard 7: What is the inverse of f(x)=x3f(x)=x^3?

Answer: f−1(x)=x3f^{-1}(x)=\sqrt[3]{x}. Cubing and cube root are inverse operations.

Flashcard 8: What restriction on cc is needed to solve x+1x−1=c\frac{x+1}{x-1}=c for xx?

Answer: c≠1c\neq 1. Denominator cannot equal zero in rational function.

Flashcard 9: What is the inverse of f(x)=x+1x−1f(x)=\frac{x+1}{x-1} with x≠1x\neq 1?

Answer: f−1(x)=x+1x−1f^{-1}(x)=\frac{x+1}{x-1} with x≠1x\neq 1. This rational function is its own inverse.

Flashcard 10: What is the solution for xx in f(x)=cf(x)=c if f(x)=x+1x−1f(x)=\frac{x+1}{x-1} and c≠1c\neq 1?

Answer: x=c+1c−1x=\frac{c+1}{c-1}. Cross-multiply and solve for xx.

Flashcard 11: What is the inverse of f(x)=5−x3f(x)=\frac{5-x}{3}?

Answer: f−1(x)=5−3xf^{-1}(x)=5-3x. Subtract from 5, then divide by 3.

Flashcard 12: What is the domain of f−1(x)f^{-1}(x) if f(x)=x+1x−1f(x)=\frac{x+1}{x-1}?

Answer: x≠1x\neq 1. Domain of inverse equals range of original function.

Flashcard 13: What is the inverse of f(x)=(x−3)3f(x)=(x-3)^3?

Answer: f−1(x)=x3+3f^{-1}(x)=\sqrt[3]{x}+3. Take cube root, then add 3.

Flashcard 14: What is the coordinate relationship between inverse functions on a graph?

Answer: Points (a,b)(a,b) on ff become (b,a)(b,a) on f−1f^{-1}. Coordinates are reflected across y=xy=x.

Flashcard 15: What test is commonly used to check if a graph represents a one-to-one function?

Answer: The horizontal line test. Each horizontal line intersects the graph at most once.

Flashcard 16: What is the inverse of f(x)=xx+1f(x)=\frac{x}{x+1} with x≠−1x\neq -1?

Answer: f−1(x)=x1−xf^{-1}(x)=\frac{x}{1-x} with x≠1x\neq 1. Cross-multiply and solve for yy after swapping variables.

Flashcard 17: What is the inverse of the function f(x)=x−af(x)=x-a?

Answer: f−1(x)=x+af^{-1}(x)=x+a. Addition and subtraction are inverse operations.

Flashcard 18: What is the inverse of f(x)=3xf(x)=\frac{3}{x} with x≠0x\neq 0?

Answer: f−1(x)=3xf^{-1}(x)=\frac{3}{x} with x≠0x\neq 0. Reciprocal function with constant numerator.

Flashcard 19: What is the inverse of f(x)=4x3f(x)=\sqrt[3]{4x}?

Answer: f−1(x)=x34f^{-1}(x)=\frac{x^3}{4}. Cube the input, then divide by 4.

Flashcard 20: What is the inverse of f(x)=2x3f(x)=2x^3?

Answer: f−1(x)=x23f^{-1}(x)=\sqrt[3]{\frac{x}{2}}. Divide by 2, then take cube root.

Flashcard 21: What is the inverse of f(x)=3x−7f(x)=3x-7?

Answer: f−1(x)=x+73f^{-1}(x)=\frac{x+7}{3}. Add 7, then divide by 3.

Flashcard 22: What is the inverse of f(x)=x3−1f(x)=x^3-1?

Answer: f−1(x)=x+13f^{-1}(x)=\sqrt[3]{x+1}. Add 1, then take cube root.

Flashcard 23: What is the key step to start finding f−1(x)f^{-1}(x) from y=f(x)y=f(x)?

Answer: Swap xx and yy in y=f(x)y=f(x). This is the first step in the standard algebraic method.

Flashcard 24: What is the inverse of f(x)=x−12x+3f(x)=\frac{x-1}{2x+3} with x≠−32x\neq -\frac{3}{2}?

Answer: f−1(x)=1+3x1−2xf^{-1}(x)=\frac{1+3x}{1-2x} with x≠12x\neq \frac{1}{2}. Cross-multiply and solve for yy after swapping variables.

Flashcard 25: What is the inverse of the function f(x)=axf(x)=ax where a≠0a\neq 0?

Answer: f−1(x)=xaf^{-1}(x)=\frac{x}{a}. Multiplication and division are inverse operations.

Flashcard 26: What is f−1(x)f^{-1}(x) if f(x)=x−7x+2f(x)=\frac{x-7}{x+2} with x≠−2x\neq -2?

Answer: f−1(x)=2x+71−xf^{-1}(x)=\frac{2x+7}{1-x} with x≠1x\neq 1. Cross-multiply and solve for yy after swapping variables.

Flashcard 27: What is the inverse of f(x)=ax+bf(x)=ax+b where a≠0a\neq 0?

Answer: f−1(x)=x−baf^{-1}(x)=\frac{x-b}{a}. Apply inverse operations in reverse order.

Flashcard 28: What is the solution for xx in f(x)=cf(x)=c if f(x)=x3−4f(x)=x^3-4?

Answer: x=c+43x=\sqrt[3]{c+4}. Add 4, then take cube root.

Flashcard 29: What must you do after swapping xx and yy to find f−1(x)f^{-1}(x)?

Answer: Solve the new equation for yy. This isolates yy to express the inverse function.

Flashcard 30: What is the range of f−1(x)f^{-1}(x) if f(x)=x+1x−1f(x)=\frac{x+1}{x-1}?

Answer: y≠1y\neq 1. Range of inverse equals domain of original function.

Flashcard 31: What is the inverse of f(x)=xnf(x)=\sqrt[n]{x} for odd nn?

Answer: f−1(x)=xnf^{-1}(x)=x^n. Odd roots and powers are inverse operations.

Flashcard 32: What is the inverse of f(x)=−2x3+6f(x)=-2x^3+6?

Answer: f−1(x)=6−x23f^{-1}(x)=\sqrt[3]{\frac{6-x}{2}}. Subtract from 6, divide by 2, then take cube root.

Flashcard 33: What is the inverse of the function f(x)=x+af(x)=x+a?

Answer: f−1(x)=x−af^{-1}(x)=x-a. Addition and subtraction are inverse operations.

Flashcard 34: What is the inverse of f(x)=2x+3x−4f(x)=\frac{2x+3}{x-4} with x≠4x\neq 4?

Answer: f−1(x)=4x+3x−2f^{-1}(x)=\frac{4x+3}{x-2} with x≠2x\neq 2. Cross-multiply and solve for yy after swapping variables.

Flashcard 35: What is the inverse of f(x)=x5f(x)=x^5?

Answer: f−1(x)=x5f^{-1}(x)=\sqrt[5]{x}. Fifth power and fifth root are inverse operations.

Flashcard 36: What is the inverse of f(x)=x+83f(x)=\sqrt[3]{x+8}?

Answer: f−1(x)=x3−8f^{-1}(x)=x^3-8. Cube the input, then subtract 8.

Flashcard 37: What is the inverse of f(x)=2x3+1f(x)=\frac{2x}{3}+1?

Answer: f−1(x)=32(x−1)f^{-1}(x)=\frac{3}{2}(x-1). Subtract 1, then multiply by 32\frac{3}{2}.

Flashcard 38: What is f−1(x)f^{-1}(x) if f(x)=x+3x−2f(x)=\frac{x+3}{x-2} with x≠2x\neq 2?

Answer: f−1(x)=2x+3x−1f^{-1}(x)=\frac{2x+3}{x-1} with x≠1x\neq 1. Cross-multiply and solve for yy after swapping variables.

Flashcard 39: What is the inverse of f(x)=7−4xf(x)=7-4x?

Answer: f−1(x)=7−x4f^{-1}(x)=\frac{7-x}{4}. Subtract from 7, then divide by 4.

Flashcard 40: What is the inverse of f(x)=x2f(x)=x^2 if the domain is restricted to x≤0x\le 0?

Answer: f−1(x)=−xf^{-1}(x)=-\sqrt{x}. Square and negative square root are inverses on non-positive domain.

Flashcard 41: What is the range restriction for f(x)=x+1x−1f(x)=\frac{x+1}{x-1}?

Answer: y≠1y\neq 1. Range excludes horizontal asymptote value.

Flashcard 42: What is the inverse of f(x)=−5x+10f(x)=-5x+10?

Answer: f−1(x)=10−x5f^{-1}(x)=\frac{10-x}{5}. Subtract from 10, then divide by 5.

Flashcard 43: What condition must ff satisfy to have an inverse function on its domain?

Answer: ff must be one-to-one on its domain. Only one-to-one functions pass the horizontal line test.

Flashcard 44: What is the inverse of f(x)=x+12f(x)=\frac{x+1}{2}?

Answer: f−1(x)=2x−1f^{-1}(x)=2x-1. Multiply by 2, then subtract 1.

Flashcard 45: What is the inverse of f(x)=x3+4f(x)=x^3+4?

Answer: f−1(x)=x−43f^{-1}(x)=\sqrt[3]{x-4}. Apply cube root, then subtract 4.

Flashcard 46: What is the inverse of f(x)=x3f(x)=\sqrt[3]{x}?

Answer: f−1(x)=x3f^{-1}(x)=x^3. Cube root and cubing are inverse operations.

Flashcard 47: What is the inverse of f(x)=4x−12x+5f(x)=\frac{4x-1}{2x+5} with x≠−52x\neq -\frac{5}{2}?

Answer: f−1(x)=1+5x4−2xf^{-1}(x)=\frac{1+5x}{4-2x} with x≠2x\neq 2. Cross-multiply and solve for yy after swapping variables.

Flashcard 48: What line reflects the graph of y=f(x)y=f(x) to get the graph of y=f−1(x)y=f^{-1}(x)?

Answer: The line y=xy=x. Graphs of inverse functions are reflections across this line.

Flashcard 49: What is the inverse of f(x)=(x+2)3−5f(x)=(x+2)^3-5?

Answer: f−1(x)=x+53−2f^{-1}(x)=\sqrt[3]{x+5}-2. Add 5, take cube root, then subtract 2.

Flashcard 50: What is the inverse of f(x)=x2f(x)=x^2 if the domain is restricted to x≥0x\ge 0?

Answer: f−1(x)=xf^{-1}(x)=\sqrt{x}. Square and square root are inverses on non-negative domain.

Flashcard 51: What is the inverse of f(x)=x−2x+5f(x)=\frac{x-2}{x+5} with x≠−5x\neq -5?

Answer: f−1(x)=5x+21−xf^{-1}(x)=\frac{5x+2}{1-x} with x≠1x\neq 1. Cross-multiply and solve for yy after swapping variables.

Flashcard 52: What is the solution for xx in f(x)=cf(x)=c if f(x)=2x3f(x)=2x^3?

Answer: x=c23x=\sqrt[3]{\frac{c}{2}}. Divide cc by 2, then take cube root.

Flashcard 53: What is the inverse of f(x)=1xf(x)=\frac{1}{x} with domain x≠0x\neq 0?

Answer: f−1(x)=1xf^{-1}(x)=\frac{1}{x} with x≠0x\neq 0. Reciprocal function is its own inverse.

Flashcard 54: What is the inverse of f(x)=x−43f(x)=\frac{x-4}{3}?

Answer: f−1(x)=3x+4f^{-1}(x)=3x+4. Multiply by 3, then add 4.

Flashcard 55: What is the inverse of f(x)=xf(x)=\sqrt{x} with domain x≥0x\ge 0?

Answer: f−1(x)=x2f^{-1}(x)=x^2 with domain x≥0x\ge 0. Square root and square are inverse operations.

Flashcard 56: What is the solution for xx in f(x)=cf(x)=c if f(x)=1xf(x)=\frac{1}{x} and c≠0c\neq 0?

Answer: x=1cx=\frac{1}{c}. Take reciprocal of cc.

Flashcard 57: What is the relationship between the domains and ranges of ff and f−1f^{-1}?

Answer: Domain of f−1f^{-1} = range of ff, and vice versa. Input and output sets are swapped between inverse functions.