Intercepts of Rational Functions

Problem 270: For the function

$f(x) = \frac{8x-3}{x^2 + 12x + 35}.$

Find the $y$ – and $x$ – intercepts.

Solution: To find the $y$ intercept, we let $x = 0$ .

$\begin{align*} f(x) = y & = \frac{8x-3}{x^2 + 12x + 35} \\ & = \frac{8(0)-3}{(0)^2 + 12(0) + 35} \\ & = \frac{-3}{7(5)} \\ & = -\frac{3}{35}. \end{align*}$

That is, the $y$ -intercept is $(0, -3/35)$ .

To find the $x$ intercept, we let $y = 0$ .

$\begin{align*} \frac{8x-3}{(x + 7)(x + 5)} = 0 \implies 8x - 3 = 0 \implies x = \frac{3}{8}\end{align*}$

Therefore, the $x$ -intercept is $(3/8, 0)$ .

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