Domanda del libro di testo
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
a. y = e^(-x)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.1.37a
Find the inverse of the function f(x)=mx, where m is a constant different from zero.
Guida verificata passo dopo passo1
Start with the given function: $f(x) = mx$, where \(m \neq 0\).
To find the inverse function, replace \(f(x)\) with \(y\): $y = mx$.
Swap the roles of \(x\) and \(y\) to find the inverse: $x = my$.
Solve this equation for \(y\) by dividing both sides by \(m\): \(y = \frac{x}{m}\).
Rewrite \(y\) as the inverse function notation: \(f^{-1}(x) = \frac{x}{m}\).
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Definition of an Inverse Function
An inverse function reverses the effect of the original function, mapping outputs back to their inputs. 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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4:03
Inverse Sine
One-to-One Functions and Invertibility
A function must be one-to-one (injective) to have an inverse, ensuring each output corresponds to exactly one input. For f(x) = mx with m ≠ 0, the function is linear and strictly monotonic, guaranteeing it is invertible.
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05:50One-Sided Limits
Finding the Inverse of a Linear Function
To find the inverse of f(x) = mx, solve the equation y = mx for x in terms of y. This involves isolating x, resulting in x = y/m, which defines the inverse function f⁻¹(x) = x/m.
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Percorso guidato
07:17Linearization
Pratica correlata
Domanda del libro di testo
75. a. Find the open intervals on which the function is increasing and decreasing.
g(x) = x(ln x)²
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Domanda del libro di testo
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + 4x
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Domanda del libro di testo
a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
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Domanda del libro di testo
88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)
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Domanda del libro di testo
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)
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