Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter P - P.1 - Graphs and Models - Exercises - Page 8: 50

Answer

The intercept with $y$ axis is $10$. There is no intercept with $x$ axis. The graph is symmetric. The plot in on the figure.
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Work Step by Step

To get the intercept with $y$ axis we put $x=0$ and solve for $y$: $$y=\frac{10}{0^2 +1} = \frac{10}{1}= 10.$$ To get the intercept with $x$ axis we put $y=0$ and solve for $x$: $$0=\frac{10}{x^2+1}.$$ Multiplying both sides by $x^2+1$ we get $$10=0$$ which is clearly not possible so there is no solution for $x$ i.e. no intercept with $x$ axis. To test for symmetry we put $-x$ instead of $x$: $$y(-x) = \frac{10}{(-x)^2+1} = \frac{10}{x^2+1} = y(x)$$ where we used $(-x)^2 = x^2$ for any number $x$. Since $y(x) = y(-x)$ the graph is symmetric with respect to the $y$ axis.
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