Question

let $f(x) = \frac{x + 6}{x^2 + 1}$.
a) find the $y$-intercept(s) of $f(x)$. list your answers as points in the form $(a, b)$. separate multiple intercepts with commas if necessary.
b) find the $x$-intercept(s) of $f(x)$. list your answers as points in the form $(a, b)$. separate multiple intercepts with commas if necessary.

Explanation:

Part (a) - y-intercept

Step1: Recall y-intercept definition

To find the \( y \)-intercept, we set \( x = 0 \) in the function \( f(x) \) and solve for \( f(0) \). The \( y \)-intercept is the point \( (0, f(0)) \).

Step2: Substitute \( x = 0 \) into \( f(x) \)

Given \( f(x)=\frac{x + 6}{x^2+1} \), substitute \( x = 0 \):

$$ f(0)=\frac{0 + 6}{0^2+1}=\frac{6}{1}=6 $$

Step1: Recall x-intercept definition

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

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

Set \( \frac{x + 6}{x^2+1}=0 \). A fraction is zero when its numerator is zero (and the denominator is not zero).

• Numerator: \( x + 6 = 0 \) gives \( x=-6 \).
• Denominator: For \( x=-6 \), \( x^2 + 1=(-6)^2+1 = 36 + 1=37

eq0 \), so the solution is valid.

Answer:

\((0, 6)\)

Part (b) - x-intercept