how to find range and domain of a function

Domain and Range of Rational Functions

In this commodity, we will learn how to detect the domain and range of a rational role either by using its graph or past using definite algebraic rules. We will offset past doing a brief review of what domain and range hateful.

So, nosotros volition look at the methods used to find the domain and range of rational functions. Finally, we will await at some examples with answers that illustrate the utilise of these methods.

ALGEBRA

Relevant for…

Learning most the domain and range of rational functions.

See examples

ALGEBRA

Relevant for…

Learning about the domain and range of rational functions.

See examples

What is the domain and range of functions?

If we remember of a function every bit a mapping diagram that relates an input to an output, the domain would exist the set of inputs and the range would exist the set of outputs. Allow'southward consider the post-obit mapping diagram:

We can see the inputs on the left and the outputs on the right. Here, the domain is the set up {2, 4, 6} and the range is the set {1, 5, ix}. If we consider the office $f(x)= 2x-3$ with the domain {one, 2, three, 4}, we tin can find the range by substituting each of the domain values in the office:

$f(1)=2(1)-3=-1$

$f(2)=2(2)-3=1$

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

$f(4)=2(4)-3=9$

Therefore, the range is the set {-ane, 1, 3, 9}.

At present, let's consider the role $f(x)=2x-1$ with domain $x \in R$ (it means that10 belongs to the fix of all real numbers). It may exist helpful to look at your graph to make it easier to decide the range:

We tin see that the graph is a straight line and every existent input has a real output, and since the line continues towards positive and negative infinity, any real number of output is possible. Thus, the range is all real numbers.

If we at present look at the quadratic function $f(x)={{x}^2}$ , which has a domain that is all existent numbers. We can expect at its graph to determine the range:

We encounter that for each input, the output is always positive. Therefore, the range of the function is equal to all real numbers greater than or equal to goose egg.

How to detect the domain and range of rational functions?

Generally, we tend to define the domain and range of functions on real numbers, then we volition do the aforementioned here. Notwithstanding, we volition use different strategies to find the domain and range of rational functions since obtaining the graphs of these functions is not very easy. Let'southward consider the following rational function:

$f(x)= \frac{3}{x + 2}$

We find that, for the input of -2, we obtain an output of $\frac{3}{0}$ .

We know that partition by cypher is undefined, so the function is undefined at this bespeak. However, any other input will have a existent number output, so nosotros conclude that the domain is all real numbers excluding -ii. We can write this every bit:

$R- \{- 2\}$

By considering the nature of the function, nosotros can see that any real number of outputs tin can be achieved with the exception of zero. This is because as10 gets larger in magnitude, the output gets smaller, only the output can never equal zilch. Therefore, the range of the role is:

$R - \{0 \}$

In full general, nosotros tin summate the domain of a rational function by identifying whatever bespeak where the function is undefined. This means finding any point that makes the denominator equal to zero.

To find the range of a rational role, nosotros tin identify any signal that cannot be reached with any input. This is generally found by considering the limits of the function as the magnitude of the inputs gets larger.

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Examples of domain and range of rational functions

The following examples illustrate the concepts detailed above.

EXAMPLE one

Discover the domain and range of the function $f(x) = - \frac{1}{x-4}$ .

graph of rational function with asymptote

Solution: Looking at the graph, it appears that the domain is $R - \{4 \}$ and the range is $R - \{0 \}$ . However, let's check this algebraically.

We know that a role is undefined for inputs that result in denominators equal to zero. We can form an equation with the denominator to find the undefined betoken:

$x-4 = 0$

$x = 4$

This confirms that the domain is $R - \{4 \}$ . To confirm the range, we take to identify the values that cannot exist reached with the given domain. Asx gets larger, the output tends to cypher, simply never becomes naught. Therefore, the range is $R - \{0 \}$ .

EXAMPLE ii

Find the domain and range of the role $f(x)= \frac{1}{x-10}$ .

Solution: In this case we practise not have a graph, so we take to solve the problem algebraically.

Again, we know that the expression $\frac{1}{0}$ is undefined, and then we form an equation with the denominator to find the undefined signal:

$x-10=0$

$x = 10$

Therefore, nosotros know that the domain of the function is $R- \{10 \}$ . To place the range, we have to identify the values that cannot exist reached with the given domain. Every bitx gets larger, the output tends to zero, but never becomes zero. Therefore, the range is $R - \{0 \}$ .

Case 3

Find the domain of the function $f(x)= \frac{3}{x-3} + \frac{1}{x+4}$ .

Solution: We know that rational functions are only divers when their denominator is different from zero. In this function, we can see that at that place are two points for which it is not divers: when $x-3 = 0$ and when $x+4 = 0$ .

This means that the part is undefined when $x = -4$ and $x = 3$ . Therefore, the domain of the role is all real numbers except -iv and 3, denoted $R - \{- 4, 3 \}$ .

If nosotros wanted to find the range of this function, despite the fact that each of the rational expressions cannot take a value of zero, in that location is an input ofx that results in zero, which is $x = - \frac{9 }{4}$ . Therefore, the range of the part is all real numbers (R).

Instance iv

Determine the domain of the role $f(x)= \frac{{{x}^2}+5}{5{{x}^3}+ 50x}$ .

Solution: Nosotros remember that rational functions are merely defined when their denominator is dissimilar from zero. Therefore, to find the domain of the function, we accept to notice the zeros of the equation in the denominator:

$5{{x}^3}+50x=0$

We can factor theten:

$x(5{{x}^2}+50)=0$

We tin can run into that we have a nada when $x=0$ . Even so, the quadratic $5{{x}^2}+50$ has no real roots. Therefore, the only zero in the denominator is $x=0$ . The domain is all existent numbers except zero, denoted $R- \{0 \}$ .