Question

Find the intercepts of the graph of the equation$y=\frac{x^2-1}{x^2-4}$.

Short answer

The intercepts for the function $y=\frac{x^2-1}{x^2-4}$are,

1. The $x$-intercepts are, $x=\pm 1$.
2. The $y$- intercept is, $y=\frac{1}{4}$.

Step-by-step solution

Step 1 Given equation is,

$y=\frac{x^2-1}{x^2-4}$

Substitute $f\left(x\right)=0$in the given equation.

$$\begin{gathered} 0=\frac{x^2-1}{x^2-4} \\ {x}^2-1=0 \\ x=\pm 1 \end{gathered}$$

Step 2 Substitute src="data:image/svg+xml;base64,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" role="math" localid="1646296172384" x=0 in the given equation.

$$\begin{gathered} y=\frac{0-1}{0-4} \\ y=\frac{1}{4} \end{gathered}$$