Determine zeros of a function

A zero of a function is an input in its domain for which $f(x)=0$ , corresponding to the $x$ -coordinate of an $x$ -intercept and, for equations, a solution to $f(x)=0$ . Learners determine and interpret real zeros from equations, graphs, tables, or numerical approximations, using methods such as factoring and checking domain restrictions, while distinguishing zeros from $y$ -intercepts and recognizing that a function may have none, one, or several; complex zeros and advanced general root theory are outside this scope.

Detailed Explanation: 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 $f(0)=6$ .

Learn by doing: Determine zeros of a function

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Linear Equations - Find X Intercept (Integer) - Standard Form

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