Zeros of a function

The zeros of a function, also referred to as roots or x-intercepts, occur at x-values where the value of the function is 0 (f(x) = 0). The zero of a function can be thought of as the input value(s) that results in an output of 0. It is worth noting that not all functions have real zeros.

Example

Find the zeros of f(x) = x + 5:

Set f(x) equal to 0:

x + 5 = 0

x = -5

There is a zero at x = -5.

Finding the zeros of a function

There are a number of different ways to find the zeros of a function, depending on the type of function. For simpler functions, it is relatively easy to set f(x) = 0 and solve for x, such as in the example above. The more complex the function, the more difficult it is to find its zeros. Factoring and the quadratic formula are two commonly used tools for finding the zeros of a function.

Solving by factoring

If a function can be factored by grouping, setting each factor equal to 0 then solving for x will yield the zeros of a function.

Example

Find the zeros of the function f(x) = x2 - 3x + 2 by factoring.

Factor:

x2 - 3x + 2 = (x - 1)(x - 2)

Set the factors equal to 0:

(x - 1)(x - 2) = 0

x - 1 = 0 and x - 2 = 0

x = 1 and x = 2

Therefore, the function x2 - 3x + 2 has zeros at x = 1, 2

Solving using the quadratic formula

Quadratic equations are studied fairly thoroughly in algebra, in part due to the prevalence of complex equations that can be reduced to quadratic equations. Since any quadratic equation can be solved using the quadratic formula, it is possible to find the zero of any quadratic equation by putting it in the form of the general quadratic equation,

ax2 + bx + c = 0,

then using the quadratic formula.

Example

Find the zeros of 2x(x - 4) = -4 using the quadratic formula.

Get f(x) into the general form of a quadratic equation:

f(x) = 2x2 - 8x + 4 = 0

Plug it into the quadratic formula:

Therefore f(x) has zeros at x = 2 + √2 and 2 - √2.

Zeros on a graph

On a graph, the zeros of a function occur when the curve crosses the x-axis. Therefore, all zeros will have a coordinate of the form (x, 0).

Example

The graph below has zeros at x = -5, -1, 3