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Use the vertical line test to identify graphs in which y is a function of x.
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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 13
The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = 2x
Guida verificata passo dopo passo1
Start with the given function: \(f(x) = 2x\). To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\), so you have \(y = 2x\).
Next, interchange the variables \(x\) and \(y\) to reflect the inverse relationship. This gives \(x = 2y\).
Now, solve this equation for \(y\) to express the inverse function. Divide both sides by 2 to isolate \(y\): \(y = \frac{x}{2}\).
Rewrite \(y\) as \(f^{-1}(x)\) to get the inverse function: \(f^{-1}(x) = \frac{x}{2}\).
To verify the inverse, compute \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). Substitute \(f^{-1}(x)\) into \(f(x)\) and simplify, then substitute \(f(x)\) into \(f^{-1}(x)\) and simplify. Both should simplify to \(x\) if the inverse is correct.
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One-to-One Functions
A one-to-one function assigns each input a unique output and vice versa, ensuring that no two different inputs produce the same output. This property is essential for a function to have an inverse, as the inverse must also be a function.
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4:07
Decomposition of Functions
Inverse Functions
The inverse of a function reverses the roles of inputs and outputs, effectively 'undoing' the original function. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, meaning applying one after the other returns the original value.
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4:30Graphing Logarithmic Functions
Verification of Inverse Functions
To verify that two functions are inverses, you must show that composing them in both orders returns the input variable. Specifically, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x must hold true for all x in the domains of the respective functions.
Video consigliato:
4:30Graphing Logarithmic Functions
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