Question

determine the $x$-intercept(s) of the following function. enter your answers as points in the form $(a, b)$. separate multiple $x$-intercepts with a comma. if there is no $x$-intercept, enter none. $f(x) = \frac{x + 4}{x^2 + 1}$

Explanation:

Step1: Recall x-intercept definition

To find the \(x\)-intercept, set \(f(x) = 0\) and solve for \(x\). The \(x\)-intercept is the point \((x, 0)\) where the function crosses the \(x\)-axis.

Step2: Set \(f(x) = 0\)

Given \(f(x)=\frac{x + 4}{x^{2}+1}\), set \(\frac{x + 4}{x^{2}+1}=0\).

Step3: Solve the equation

A fraction is zero when its numerator is zero (and the denominator is not zero). So, solve \(x + 4=0\) and check \(x^{2}+1
eq0\).
From \(x + 4 = 0\), we get \(x=-4\).
Now, check the denominator: For \(x=-4\), \(x^{2}+1=(-4)^{2}+1 = 16 + 1=17
eq0\). So \(x = - 4\) is a valid solution.

Answer:

\((-4,0)\)