Domanda del libro di testo
In Exercises 31–32, the domain of each piecewise function is (-∞, ∞) (a) Graph each function. (b) Use the graph to determine the function's range.
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 31
Which graphs in Exercises 29–34 represent functions that have inverse functions?
Guida verificata passo dopo passo1
Recall that a function has an inverse function if and only if it is one-to-one, meaning each output corresponds to exactly one input.
Use the Horizontal Line Test on each graph: if any horizontal line intersects the graph more than once, the graph does not represent a one-to-one function and therefore does not have an inverse function.
For each graph, imagine drawing horizontal lines across the entire domain and check how many times each line intersects the graph.
If every horizontal line intersects the graph at most once, then the function is one-to-one and has an inverse function.
Summarize which graphs pass the Horizontal Line Test and thus represent functions with inverse functions.
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Definition of a Function
A function is a relation where each input corresponds to exactly one output. Understanding this ensures that when analyzing graphs, each x-value has only one y-value, which is essential before considering inverses.
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5:57
Graphs of Common Functions
One-to-One Functions
A function has an inverse only if it is one-to-one, meaning each output corresponds to exactly one input. This property ensures the inverse relation is also a function, which is critical when identifying invertible graphs.
Video consigliato:
4:07Decomposition of Functions
Horizontal Line Test
The horizontal line test is a graphical method to determine if a function is one-to-one. If any horizontal line intersects the graph more than once, the function fails the test and does not have an inverse function.
Video consigliato:
06:49The Slope of a Line
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