Domain And Range Worksheet Answers

The domain can be calculated past finding the set of all possible values for the independent variable, ordinarilyten. The range can be calculated by finding the gear up of all possible values for the dependent variable, generallyy.

Here, nosotros volition wait at a summary of the domain and range of a function. Nosotros will also review how to detect the domain and range. In addition, we will solve several domain and range exercises to larn the reasoning used when solving these types of exercises.

ALGEBRA

domain and range of an irrational function

Relevant for…

Finding the domain and range of functions with examples.

See examples

ALGEBRA

Relevant for…

Finding the domain and range of functions with examples.

See examples

Summary of domain and range

Domain

The domain of a function is the fix of all possible values of the independent variable. That is, the domain is the fix of all values ofx that volition cause the office to produce existent values ofy.

To notice the domain of a function, we call back the following:

The denominator of a fraction cannot be equal to cypher.
The number nether a square root sign cannot exist negative.

Therefore, the domain of a office is establish by looking for the values of the independent variable (usuallyx) that we can use. We have to avert 0 in the denominator of a fraction and negative numbers in foursquare roots.

Range

The range of a part is the set of all possible values of the dependent variable (usuallyy) after using the domain. This ways that the range is the set of the values ofy that we get after using all the values ofx.

To find the range of a function, nosotros consider the post-obit:

The range is the extent of they values, from the minimum value to the maximum value.
We can substitute values ofx to analyze the result, considering whether the values are negative or positive.
Nosotros have to brand sure nosotros are looking for maximum and minimum values.

Domain and range – Examples with answers

The following domain and range examples accept their respective solution. Each solution details the process and reasoning used to obtain the answer.

Case 1

Detect the domain and the range of the role $latex f(x)={{x}^ii}+1$.

Solution

Domain:The function $latex f(10)={{x}^2}+1$ is defined for all existent values ofx considering there are no restrictions of the values ofx. Therefore, the domain ofx is:

"All existent values ofx"

Range:Given that $latex {{x}^2}$ is never negative, $latex {{ten}^ii}+1$ is never less than 1. Therefore, the range of $latex f(x)$ is:

"All real values of $latex f(x) \geq 1$"

Nosotros tin see thatx can accept whatever value in the graph, but the values of $latex y=f(x)$ are greater than or equal to one.

Instance 2

Discover the domain and the range of the office $latex f(x)= \frac{1}{x+three}$.

Solution

Domain:The function $latex f(10)= \frac{ane}{x+iii}$ is not defined for $latex x=-3$ since this would produce a division past zero (nosotros would have a 0 in the denominator). Therefore, the domain of $latex f(10)$ is:

"All real numbers except the -3"

Range:No matter how large or minorx is, the values of $latex f(10)$ will never equal aught. If we attempt to solve the equation for 0, we have:

$latex 0= \frac{i}{x+iii}$

Multiplying both sides by $latex x + three$, we take:

$latex 0= 1$

This is impossible. Therefore, the range of $latex f(10)$ is:

"All real numbers except zero"

In the graph, nosotros can see that the function is not divers for $latex x = -3$ and that the role takes ally values except zero:

EXAMPLE 3

Discover the domain and the range of the part $latex f(t)=\sqrt{5-t}$.

Solution

Domain:The part $latex f(t)=\sqrt{5-t}$ is not defined for existent numbers greater than 5 since this would result in negative numbers beneath the foursquare root and imaginary numbers for $latex f(t)$, so the domain of $latex f(t)$ is:

"All existent numbers, $latex t\leq 5$"

Range:By definition of a square root we accept:

$latex f(t)=\sqrt{5-t}\geq 0$

Therefore, the range of $latex f(t)$ is:

"All real numbers, $latex f(t)\geq 0$"

In the graph, we can see that $latex t$ does not take values greater than 5 and that the range is greater than or equal to 0:

EXAMPLE iv

Determine the domain and the range of the role $latex f(x)=\frac{{{x}^2}+x-two}{{{x}^2}-x-ii}$.

Solution

Domain:The simply problem we have with this function is that we have to be careful not to dissever by zero. Then the values of10 cannot have those values that produce a segmentation by zero. Therefore, nosotros make the denominator equal to nada and solve:

$latex {{10}^2}-x-2=0$

$latex (x+1)(x-2)=0$

$latex 10=-ane$ o $latex x=2$

Thus, the domain is:

"All existent numbers except -1 and 2"

Range:The range is a flake more difficult to determine, but in this case, nosotros have no visible constraints that brand the range greater or less than a specific value. Nosotros cheque this with the graph of the function:

We see that the office takes all the values ofy, so the range is:

"All real numbers"

Example v

Determine the domain and the range of the role $latex chiliad(10)=-\sqrt{-2x+five}$.

Solution

Domain:The only problem we have with this part is that we cannot have negative values inside the square root sign. Therefore, we can brand the expression inside the square root greater than or equal to zero and solve:

$latex -2x+five\geq 0$

$latex -2x\geq -five$

$latex 2x\leq 5$

$latex x\leq \frac{five}{two}$

Therefore, the domain is:

"All existent numbers $latex 10\leq \frac{5}{2}$"

Range:We know that past definition, the result of a square root must be greater than or equal to nothing. Yet, in this case, nosotros accept a negative sign that precedes the foursquare root, so the range is:

"All real numbers $latex g(x)\leq 0$"

We can look at the graph to check this:

Instance 6

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

Solution

Domain:This is an easy problem. There are no denominators, so we don't have division past zero bug and there are no radicals, so we don't take issues with negative numbers in square roots. When nosotros have a polynomial like in this case, the domain is:

"All real numbers"

Range:The range varies from polynomial to polynomial. In this case, we know that $latex {{x}^4} $ is always positive, so when preceded by a negative sign, we always obtain a negative number or equal to 0. That ways that the highest point is 2 and the range is:

All existent numbers, $latex f(x)\leq ii$"

In the graph, nosotros can come across this: