Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.1.37a

Chapter 7, Problem 7.1.37a

Find the inverse of the function f(x)=mx, where m is a constant different from zero.

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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