Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 0 - Before Calculus - 0.1 Functions - Exercises Set 0.1 - Page 15: 35

Answer

(i) The rational function has factors $x+2$ and $x-1$ which can be cancelled. (ii) $g(x) = x + 1$

Work Step by Step

(i) Restrictions on a rational function include any $x$ that makes its denominator equal to zero. The function's graph will have a hole in its graph at that $x$ if that factor can be cancelled. $f(x) = \frac{(x + 2)(x^2 - 1)}{(x + 2)(x - 1)}$ $= \frac{(x + 2)(x - 1)(x + 1)}{(x + 2)(x - 1)} = x+1, x \ne -2, 1$ with holes at $x = -2$ and $x = 1$ since both factors can be cancelled. (ii) A function whose graph is identical to that of $f$ but without the holes will therefore be $g(x) = x + 1$ for all values of $x$, since this is the reduced form of $f$ without the restricted/cancelled factors
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