Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Function Composition
2:09 minutes
Problem 21d
Textbook Question
Textbook QuestionDetermine whether each function graphed or defined is one-to-one. y = 2x^3 - 1
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A function is considered one-to-one if it assigns a unique output for every unique input, meaning no two different inputs produce the same output. This can be visually assessed using the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.
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Cubic Functions
Cubic functions are polynomial functions of degree three, typically expressed in the form y = ax^3 + bx^2 + cx + d. These functions can exhibit various behaviors, including having one or two turning points, which can affect their one-to-one nature. The specific function given, y = 2x^3 - 1, is a cubic function that generally increases without bound.
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Graphical Analysis
Graphical analysis involves examining the shape and behavior of a function's graph to derive insights about its properties. For the function y = 2x^3 - 1, analyzing its graph can help determine if it is one-to-one by observing whether any horizontal lines intersect the graph more than once, which would indicate that it is not one-to-one.
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