To find the x- and y-intercepts of the function:

$f(x) = \frac{x + 5}{x^2 + 4}$

1. x-intercept:

The x-intercept occurs when $f(x) = 0$, or equivalently, when $y = 0$. This means we set the numerator equal to zero and solve for x.

$x + 5 = 0$ $x = -5$

So, the x-intercept is:

$(x, y) = (-5, 0)$

2. y-intercept:

The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the function:

$f(0) = \frac{0 + 5}{0^2 + 4} = \frac{5}{4}$

So, the y-intercept is:

$(x, y) = (0, \frac{5}{4})$

Summary:

• x-intercept: $(-5, 0)$
• y-intercept: $(0, \frac{5}{4})$

Would you like more details or have any further questions?

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Here are 5 related questions:

1. How do you find intercepts for more complex functions?
2. Can a function have more than one x-intercept or y-intercept?
3. What happens if the numerator has no real solutions for the x-intercept?
4. How do you interpret intercepts graphically?
5. What if the denominator equals zero at a certain point?

Tip: Always check if the denominator is zero to avoid undefined values in rational functions!