# Range

## Key Questions

• The range of a function is its y-values or outputs. If you look at the graph from lowest point to highest point, that will be the range.

Ex: $y = {x}^{2}$ has a range of y$\ge$ 0 since the vertex is the lowest point, and it lies at (0,0).

Ex: y = 2x + 1 has a range from $- \setminus \infty$ to $\setminus \infty$ since the ends of the graph point in those directions. (down and left, and up and right)
In interval notation, you would write $\left(- \setminus \infty , \setminus \infty\right)$.

Ex: Some functions have interesting ranges like the sine function.
y = sin(x)
Its highest values are 1 and its lowest values are -1. That range is $- 1 \le y \le 1$ or [-1,1] in interval notation.

Ex: A rather complicated function with a very challenging range is the inverse or reciprocal function, $y = \frac{1}{x}$.

The output values might be difficult to describe except to say that they seem to include all real numbers except 0. (there is a horizontal asymptote on the x-axis)

You could write $\left(- \setminus \infty , 0\right) U \left(0 , \setminus \infty\right)$ in interval notation.

• The range of a function is the set of all possible outputs of that function.

For example, let's look at the function $y = 2 x$

Since we can plug in any x value and multiple it by 2, and since any number can be divided by 2, the output of the function, the $y$ values, can be any real number.

Therefore, the range of this function is "all real numbers"

Let's look at something slightly more complicated, a quadratic in vertex form: $y = {\left(x - 3\right)}^{2} + 4$. This parabola has a vertex at $\left(3 , 4\right)$ and opens upwards, therefore the vertex is the minimum value of the function. The function never goes below 4, therefore the range is $y \ge 4$.

• Just like describing the domain of a function, you can use inequalities or interval notation; for example, you can write:

Range: $\left[- 2 , 3\right)$ or $- 2 \le y < 3$

I hope that this was helpful.

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