Concepts I touch on that you should be familiar with:

• Domain and range of a function
• Graphs/coordinate planes

Imagine you are given this function: y = √x²

This function takes the square root of the square of a number, which sounds a lot like the square root and the exponent of 2 cancel each other out. This would leave you with simply y = x.

That function is just a line of slope 1 through the origin (0,0) of the coordinate plane, and it means that when you input any value for x, the output y value will be the exact same as x. If x is 3, y is 3. If x is -4, y is -4. If x is π (3.14159265…), then so is y.

Now try plugging in x = 3 into the initial function of y = √x².

x = 3; y = √(3)²; y = √9; y = 3

So far so good! Now try plugging in x = -4 into the equation.

x = -4; y = √(-4)²; y = √16; y = 4

Why is y suddenly positive 4? When we set x as -4 in the y = x equation, the y value we produced was negative 4, not positive 4. So is y = x not in fact equal to y = √x²?

The easiest way to visualize the difference between these two functions is to look at them on a graph.

x = y:

As you can see, every x value lines up exactly with the same y value. The domain is all real numbers (ℝ) or (-∞, ∞). The range, more importantly, is also all real numbers because each y value must match up with an x value which can be any real number.

y = √x²:

The two graphs share many similarities, but this one seems to bounce off the x-axis, never becoming negative. The domain of the function is also all real numbers, but the range is only [0,∞). Any x value in that range will produce the same y value, but if the x value is less than 0, the y value won’t match it.

Looking back at y = √x², this change of behavior makes sense. The first step in computing the y value for a given x is to square the x value. If the x value is negative, the negative disappears, because a negative multiplied by a negative is a positive. (-3)² is 9, because -3 times -3 is positive. When the square root is taken, it is always acting on a positive number, making the output always positive, no matter if x was negative.

Another thing to note is the function y = √x also has a range (and domain) of [0,∞) since there is no number that can be multiplied with itself (squared) and be negative. There is also no real solution to the square root of a negative number!

These concepts combined lead to the confusing range of y = √x². Be careful when simplifying square root functions! Things may not always be as they seem…