Domanda del libro di testo
In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16
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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 11
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) = x +3
Guida verificata passo dopo passo1
Start with the given function: \(f(x) = x + 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x + 3\).
Next, interchange the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = y + 3\).
Now, solve this equation for \(y\) to express the inverse function: subtract 3 from both sides to get \(y = x - 3\).
Rewrite \(y\) as \(f^{-1}(x)\) to write the inverse function explicitly: \(f^{-1}(x) = x - 3\).
To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f\): \(f(f^{-1}(x)) = (x - 3) + 3 = x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}\): \(f^{-1}(f(x)) = (x + 3) - 3 = x\). Both compositions return \(x\), confirming the inverse is correct.
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One-to-One Functions
A one-to-one function assigns each input a unique output, ensuring no two different inputs share the same output. This property is essential for a function to have an inverse, as the inverse must reverse the mapping without ambiguity.
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4:07
Decomposition of Functions
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, effectively undoing the original operation.
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4:30Graphing Logarithmic Functions
Verification of Inverse Functions
To confirm that two functions are inverses, you compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). If both compositions simplify to x, the identity function, the functions are true inverses, validating the correctness of the inverse equation.
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4:30Graphing Logarithmic Functions
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