Square Function — Definition, Formula & Examples

The square function is a function that takes a number and multiplies it by itself. If you input 3, you get 9; if you input −3, you also get 9.

The square function is defined as $f(x) = x^2$ for all real numbers $x$ . It maps each input to its second power, producing a parabola that opens upward with its vertex at the origin $(0, 0)$ .

Key Formula

f(x) = x^2

Where:

$x$ = The input value (any real number)
$f(x)$ = The output, equal to x multiplied by itself

How It Works

To evaluate the square function, take whatever input value you have and multiply it by itself. The output is always zero or positive, because a negative times a negative is positive and a positive times a positive is positive. The graph of

f(x) = x^2

is a U-shaped curve called a parabola. It is symmetric about the

y

-axis, meaning

f(-x) = f(x)

for every

x

. The function decreases on the interval

(-\infty, 0)

and increases on the interval

(0, \infty)

Worked Example

Problem: Evaluate the square function for x = −5 and x = 4.

Evaluate at x = −5: Multiply −5 by itself.

f(-5) = (-5)^2 = (-5)\times(-5) = 25

Evaluate at x = 4: Multiply 4 by itself.

f(4) = 4^2 = 4 \times 4 = 16

Answer:

f(-5) = 25

and

f(4) = 16

. Both outputs are positive.

Why It Matters

The square function is the foundation for quadratic equations, which appear throughout algebra and geometry. Calculating areas of squares, modeling projectile motion, and understanding parabolas all rely on squaring. Recognizing its shape and properties helps you graph and solve quadratics faster.

Common Mistakes

Mistake: Thinking that

(-3)^2

equals

-9

Correction: The parentheses matter:

(-3)^2 = (-3)\times(-3) = 9

. A negative times a negative is always positive. Note that

-3^2 = -(3^2) = -9

is different because the exponent applies only to 3.