Skill: Determine zeros of a function

A zero of a function is an input value $x$ that makes the function equal to zero:

$$f(x)=0.$$

On a graph, a zero is the $x$-coordinate where the graph crosses or touches the $x$-axis. It is not the same as the $y$-intercept, which occurs when $x=0$.

Example

Determine the real zeros of

$$f(x)=\frac{(x-2)(x+3)}{x-1}.$$

Step 1: Set the function equal to zero.

$$\frac{(x-2)(x+3)}{x-1}=0$$

A fraction equals zero only when its numerator is zero and its denominator is not zero. Therefore, set the numerator equal to zero:

$$(x-2)(x+3)=0.$$

Step 2: Solve each factor.

Set each factor equal to zero:

$$x-2=0 \quad \text{or} \quad x+3=0.$$

So,

$$x=2 \quad \text{or} \quad x=-3.$$

Step 3: Check the domain restriction.

The denominator cannot equal zero:

$$x-1\ne 0,$$

so $x\ne 1$. Neither $2$ nor $-3$ is excluded.

Therefore, the real zeros are

$$\boxed{x=2 \text{ and } x=-3}.$$

These correspond to the $x$-intercepts $(2,0)$ and $(-3,0)$. The $y$-intercept is different: it is found by using $x=0$, giving $f(0)=6$.