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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q22E (page 97)

Which of the functions f from R toR in Exercises 21 through 24 are invertible?22 $f (x) = 2^{x}$ .

Short Answer

Expert verified

The function $f (x) = 2^{x}$ fromR toR is invertible.

Step by step solution

01

Invertible Function

Function $f (x)$ is invertible iff is one-to-one function, that is there are no values $x_{1} 鈮� x_{2}$ such that $f (x_{1}) = f (x_{2})$ .If we can find such values $x_{1}, x_{2}$ so that $x_{1} 鈮� x_{2}$ and $f (x_{1}) = f (x_{2})$ then $f (x)$ is not invertible.

02

Finding such values such that x1≠x2 but f(x1)=f(x2)

Let $y = f (x)$ and solve for $x : y = 2^{x}$ , $x = \log_{2} y$ . This means that for every y there is unique x so that $f (x) = y$ , so $f (x)$ is invertible.

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