Quadratic functions are some of the relationships that we will always find in the study of algebra. One of the important characteristics of these functions is their domain and range. Here, we will start with a quick review of what the domain and range of a function represent.

Then, we will learn how to find the domain and range of quadratic functions. Also, we will look at several examples with answers to master this topic completely.

## Summary of domain and range

### Domain of a function

The domain of a function is the set of all possible values of the independent variable, which are commonly known as the values of *x*. We can find the domain by identifying particular values of *x* that cause the function to show “improper” behavior and we have to exclude those values.

To determine the domain, we especially look for values of *x* that make the denominator zero since we cannot have division by zero and values that result in having negative numbers within square roots.

### Range of a function

The range of a function is the set of all the output values that are obtained after using the values of *x* in the domain. This means that we need to find the domain first to describe the range. The range is commonly known as the value of *y*.

Finding the range is a bit more difficult than finding the domain. To facilitate this, it is advisable to graph the function with a graphing calculator or to try to obtain a basic graph of the function by hand.

It is important to have an idea of what the graph will look like in order to describe the range of the function correctly.

## How to find the domain and range of quadratic functions?

The domain of quadratic functions can be found by determining which values of *x* we can use and which we cannot. Specifically, we must avoid values of *x* that cause the function to have zero on the denominators since they would result in division by zero.

Also, we must avoid values of *x* that cause us to have negative values within the square root or other odd roots. In the case of quadratic functions, we have neither denominators nor square roots, so we have no restrictions with the domain.

That means that the domain is equal to all real numbers in *x*. In set notation this is represented as:

$latex \{x | x \in R \}$

In interval notation this is represented as:

$latex (- \infty, + \infty)$

We know that the graphs of quadratic functions have maximums or minimums. Therefore, to find the range of a quadratic function, we have to determine its maximum or minimum point. This can be easily found by making a basic graph of the function.

Alternatively, the range can be found by algebraically by determining the vertex of the graph of the function and determining whether the graph opens up or down. The graph opens up if the coefficient of the quadratic term is positive and it opens down if the coefficient of the quadratic term is negative.

## Domain and range of quadratic functions – Examples with answers

The following examples can be used to understand the process applied to find the domain and range of quadratic functions. Try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

Find the domain and the range of the function $latex f(x)={{x}^2}+2$.

##### Solution

By using several values of *x*, we can verify that we are not restricted to any values, so the domain of this function is all real numbers.

Determining the range is a bit more complicated. We can draw a graph of this function to see what its minimum or maximum point is:

In the graph, we can see that the minimum value of the range is $latex y=2$ and the function has values greater than that. This means that the range of the function is $latex y \geq 2$.

**EXAMPLE 2**

Find the domain and the range of the quadratic function: $latex f(x)=-{{x}^2}+4$.

##### Solution

The domain of this function is all real numbers since there are no values of *x* that will cause the function to produce “inappropriate” values.

To determine the domain, we sketch a basic graph of this function:

We see that this graph opens downward. The graph has a maximum point at $latex y = 4$ and takes all values less than $latex y = 4$. Thus, the range of this function is $latex y \leq 4$.

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**EXAMPLE 3**

Determine the domain and the range of the function $latex f(x)={{x}^2}+4x-1$.

##### Solution

From the previous exercises, we can already realize that quadratic functions have a domain that is equal to all values of *x*.

To find the range of the function, we can either graph or rewrite the function in vertex form and find the maximum or minimum point.

The function is given in the standard form $latex f(x)=a{{x}^2}+bx+c$, so now we are going to convert it to the vertex form $latex f(x)=a{{(xh)}^2}+k$, where $latex (h, k)$ is the vertex. Therefore, completing the square, we have:

$latex f(x)={{x}^2}+4x-1$

$latex =({{x}^2}+4x)-1$

$latex =({{x}^2}+4x+4)-1-4$

$latex ={{(x+2)}^2}-5$

The quadratic term of this function is positive, so its graph opens upward. This means that the vertex, which is equal to $latex (-2, -5) $, represents the minimum point. Thus, the range is $latex y \geq -5$.

**EXAMPLE 4**

Find the range of the function $latex f(x)=2{{x}^2}+12x+16$.

##### Solution

In addition to using the method of completing the square to find the vertex, we can also use formulas to find the vertex directly. The coordinates of *h* and *k* of the vertex of the graph are given by:

$latex h=\frac{-b}{2a}=\frac{-12}{2(2)}=-3$

$latex k=f(-3)=-2$

The coefficient of the quadratic term is positive, which means that the graph of the function opens upward and the vertex (-3, -2) represents the minimum point. The range of the function is $latex y≥-2$.

## Domain and range of quadratic functions – Practice problems

Put your knowledge of domain and range into practice to solve the following problems. If you need help you can look at the examples solved above.

## See also

Interested in learning more about domain and range of functions? Take a look at these pages:

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