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Chapter 1: Problem 62

Use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. $$y=\frac{4}{x^{2}+1}$$

Short Answer

Expert verified
The graph of the function \(y=\frac{4}{x^{2}+1}\) intercepts the y-axis at (0, 4) and has no x-intercept.

Step by step solution

01

Plotting the Graph

Plot the graph of the equation \(y=\frac{4}{x^{2}+1}\) using a graphing tool. A standard setting usually means an x and y range from -10 to 10. Plot the graph within this range.
02

Analyzing the Graph for Intercepts

An intercept occurs when the graph crosses either the x-axis or the y-axis. The x-intercepts are the values of x for which \(y=0\), and the y-intercept is the value of y where \(x=0\). For the given equation, there should be a y-intercept and no x-intercepts as the function does not cross the x-axis within the standard setting.
03

Approximating the intercepts

To find the y-intercept, set \(x=0\) in the equation to get the y-value. In this case, substituting \(x=0\) in the given function, we will get \(y=4\). Therefore, there is a y-intercept at (0, 4). For x-intercept, since the graph does not intersect the x-axis in the standard setting, there is no x-intercept. So, the function intercepts the y-axis at (0,4) and does not intercept the x-axis.

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