Domanda del libro di testo
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 a. f(6)
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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 27
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 +1)/(x-3)
Guida verificata passo dopo passo1
Start by writing the function as an equation with y: \(y = \frac{2x + 1}{x - 3}\).
To find the inverse, swap x and y: \(x = \frac{2y + 1}{y - 3}\).
Solve this new equation for y. Begin by multiplying both sides by \((y - 3)\) to eliminate the denominator: \(x(y - 3) = 2y + 1\).
Distribute x on the left side: $xy - 3x = 2y + 1$. Then, collect all terms involving y on one side and constants on the other: $xy - 2y = 3x + 1$.
Factor y out on the left side: \(y(x - 2) = 3x + 1\). Finally, solve for y by dividing both sides by \((x - 2)\): \(y = \frac{3x + 1}{x - 2}\). This y represents the inverse function \(f^{-1}(x)\).
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One-to-One Functions
A function is one-to-one if each output corresponds to exactly one input, meaning it passes the horizontal line test. This property ensures the function has an inverse because no two different inputs produce the same output, allowing the inverse to be well-defined.
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Decomposition of Functions
Finding the Inverse Function
To find the inverse of a function, swap the roles of x and y in the equation and solve for y. This process reverses the input-output relationship, producing a function that 'undoes' the original function's operation.
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4:30Graphing Logarithmic Functions
Verification of Inverse Functions
To verify an inverse function, compose the original function with its inverse in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x, confirming that the functions are true inverses of each other.
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4:30Graphing Logarithmic Functions
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