Domanda del libro di testo
Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2
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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 93
The functions in Exercises 93–95 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(f-1(x)) = x and f-1(f(x)) = x. f(x) = 4x - 3
Guida verificata passo dopo passo1
Start with the given function: \(f(x) = 4x - 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = 4x - 3\).
Next, swap the roles of \(x\) and \(y\) to find the inverse: replace \(y\) with \(x\) and \(x\) with \(y\), giving \(x = 4y - 3\).
Solve this new equation for \(y\) to express the inverse function: add 3 to both sides to get \(x + 3 = 4y\), then divide both sides by 4 to isolate \(y\): \(y = \frac{x + 3}{4}\).
Rewrite \(y\) as \(f^{-1}(x)\), so the inverse function is \(f^{-1}(x) = \frac{x + 3}{4}\).
To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f\): \(f\left(f^{-1}(x)\right) = 4 \times \frac{x + 3}{4} - 3\). Simplify this expression and check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}\): \(f^{-1}(4x - 3) = \frac{(4x - 3) + 3}{4}\). Simplify and check if it equals \(x\).
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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.
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Decomposition of Functions
Inverse Functions
The inverse of a function reverses the roles of inputs and outputs, denoted as f⁻¹(x). To find the inverse, solve the equation y = f(x) for x in terms of y, then interchange x and y. The inverse 'undoes' the original function.
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4:30Graphing Logarithmic Functions
Verification of Inverse Functions
To verify that two functions are inverses, show that composing them in either order returns the input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that each function reverses the effect of the other.
Video consigliato:
4:30Graphing Logarithmic Functions
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