College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 2 - Section 2.6 - Graphs of Basic Functions - 2.6 Exercises - Page 240: 41


$f(x)=\left\{\begin{array}{l} \sqrt[3]{x}~~~\mathrm{i}\mathrm{f}~x\lt 1\\ x+1~~~\mathrm{i}\mathrm{f}~x\geq 1 \end{array}\right.$ Domain: $(-\infty,\ \infty)$ Range: $(-\infty,1)\cup[2,\ \infty$)

Work Step by Step

We see that the graph consists of the cube root function $y=\sqrt[3]{x}$ from $-\infty$ until $x=1$ and the the line $y=x+1$ afterwards. Solid circles indicate that the value belongs to that piece of the function (e.g. "$\leq$" or "$\geq$"), while open circles indicate that the value does not (e.g. "$\lt$" or "$\gt$"). Thus we have: $f(x)=\left\{\begin{array}{l} \sqrt[3]{x}~~~\mathrm{i}\mathrm{f}~x\lt 1\\ x+1~~~\mathrm{i}\mathrm{f}~x\geq 1 \end{array}\right.$ We see that the domain consists of all real numbers, while the range consists of numbers smaller than 1 and greater than or equal to 2: Domain: $(-\infty,\ \infty)$ Range: $(-\infty,1)\cup[2,\ \infty$)
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