Call Now to Set Up Tutoring:

(888) 888-0446

HiSET: Math : Domain and range of a function

Study concepts, example questions & explanations for HiSET: Math

Create An Account

All HiSET: Math Resources

125 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Domain And Range Of A Function

Define functions $f(x) = \sqrt{x}$ and $g(x) =- x^{2}$

Give the domain of the composition $g\circ f$ .

Possible Answers:

$\left ( 0, \infty\right )$

$\left [ 0, \infty\right )$

$\left ( - \infty, 0 \right ]$

$\left ( - \infty, 0 \right )$

None of the other choices gives the correct response.

Correct answer:

$\left [ 0, \infty\right )$

Explanation:

The domain of the composition of functions $g\circ f$ is that the set of all values that fall in the domain of such that f(c) falls in the domain of .

$g(x) =- x^{2}$ , a polynomial function, whose domain is the set of all real numbers. Therefore, the domain is not restricted further by . The domain of $g\circ f$ is $\left [ 0, \infty\right )$ .

Report an Error

Example Question #2 : Domain And Range Of A Function

Define functions $f(x) = \frac{1}{x-1}$ and $g(x)= \sqrt{x}+ 1$ .

Give the domain of the composition $g\circ f$ .

Possible Answers:

$(0, 1) \cup (1, \infty)$

$(-\infty, 1)$

(0, 1)

$(1, \infty)$

$(-\infty, 1) \cup (1, \infty)$

Correct answer:

$(1, \infty)$

Explanation:

The domain of the composition of functions $g\circ f$ is that the set of all values that fall in the domain of such that f(c) falls in the domain of .

$f(x) = \frac{1}{x-1}$ , a rational function, so its domain is the set of all numbers except those that make its denominator zero. x-1= 0 if and only if x= 1 , making its domain $(-\infty, 1) \cup (1, \infty)$ .

$g(x)= \sqrt{x}+ 1$ , a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or $\left [ 0, \infty\right )$ .

Therefore, we need to further restrict the domain to those values in $(-\infty, 1) \cup (1, \infty)$ that make

$f(c) \ge 0$

Equivalently,

$\frac{1}{c-1}\ge 0$

Since 1 is positive, and the quotient is nonnegative as well, it follows that

c-1 > 0 ,

and

c > 1 .

The domain of $g\circ f$ is therefore $(1, \infty)$ .

Report an Error

Example Question #5 : Functions And Function Notation

Define functions $f(x) = x^{2}-16$ and $g(x) = \sqrt{x}$ .

Give the domain of the composition $g\circ f$ .

Possible Answers:

$(-\infty, -4]$

$[4 , \infty)$

[-4, 4 ]

$(-\infty, -4] \cup [4 , \infty)$

$(-\infty, \infty)$

Correct answer:

$(-\infty, -4] \cup [4 , \infty)$

Explanation:

The domain of the composition of functions $g\circ f$ is that the set of all values that fall in the domain of such that f(c) falls in the domain of .

$f(x) = x^{2}-16$ , a polynomial function, so its domain is the set of all real numbers.

$g(x) = \sqrt{x}$ , so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or $\left [ 0, \infty\right )$ .

Therefore, the domain of $g\circ f$ is the set of all real numbers such that

$f(c) \ge 0$ .

Equivalently,

$c^{2}-16 \ge 0$

To solve this, first, solve the corresponding equation:

$c^{2}-16 = 0$

Factor the polynomial at left using the difference of squares pattern:

(c+4)(c-4) = 0

By the Zero Factor Property, either

c+ 4 = 0 , in which case c= -4 ,

c- 4 = 0 , in which case c= 4 .

These are the boundary values of the intervals to be tested. Choose an interior value of each interval and test it in the inequality:

$(-\infty, -4)$

Testing c= -5 :

$(-5)^{2}-16 \ge 0$

$25-16 \ge 0$

$9 \ge 0$ - True; include

( -4, 4 )

Testing c= 0 :

$0^{2}-16 \ge 0$

$0 -16 \ge 0$

$-16 \ge 0$ - False; exclude

$( 4 , \infty)$

Testing c= 5 :

$5^{2}-16 \ge 0$

$25-16 \ge 0$

$9 \ge 0$ - True; include

Include $(-\infty, -4)$ and $( 4 , \infty)$ . Also include the boundary values, since equality is included in the inequality statement ( $\ge$ ).

The correct domain is $(-\infty, -4] \cup [4 , \infty)$ .

Report an Error

Example Question #1 : Functions And Function Notation

Define functions $f(x) = x^{2}-16$ and $g(x) = \sqrt{x}$ .

Give the domain of the composition $f \circ g$ .

Possible Answers:

$\left [ -4, \infty\right )$

$\left [ 0, \infty\right )$

[ -4,4]

$\left ( -\infty, \infty\right )$

$\left [ 4, \infty\right )$

Correct answer:

$\left [ 0, \infty\right )$

Explanation:

The domain of the composition of functions $f \circ g$ is the set of all values that fall in the domain of such that g(c) falls in the domain of .

$g(x) = \sqrt{x}$ , a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or $\left [ 0, \infty\right )$ .

$f(x) = x^{2}-16$ , a polynomial function; its domain is the set of all reals, so it does not restrict the domain any further.

The correct domain is $\left [ 0, \infty\right )$ .

Report an Error

Example Question #1 : Functions And Function Notation

Define functions $f(x) = \frac{1}{x-1}$ and $g(x)= \sqrt{x}+ 1$ .

Give the domain of the composition $f \circ g$ .

Possible Answers:

$\left (1, \infty\right )$

$\left [ 1, \infty\right )$

$\left ( -\infty, \infty\right )$

$\left [ 0, \infty\right )$

$\left ( 0, \infty\right )$

Correct answer:

$\left ( 0, \infty\right )$

Explanation:

The domain of the composition of functions $f \circ g$ is the set of all values that fall in the domain of such that g(c) falls in the domain of .

$f(x) = \frac{1}{x-1}$ , a rational function, so its domain is the set of all numbers except those that make its denominator zero. x-1= 0 if and only if x= 1 , so from $\left [ 0, \infty\right )$ , we exclude only the value so that

g(c) = 1 ;

that is,

$\sqrt{c}+ 1 = 1$

$\sqrt{c}+ 1 - 1 = 1 - 1$

$\sqrt{c} = 0$

c= 0

Excluding this value from the domain, this leaves $\left ( 0, \infty\right )$ as the domain of $f \circ g$ .

Report an Error

Example Question #11 : Functions And Function Notation

Define functions $f(x) = \sqrt{x-8}$ and $g(x) = \sqrt{x+ 8}$ .

Give the domain of the function f+g .

Possible Answers:

$[-8, \infty )$

$(-\infty , -8]$

$[8, \infty )$

$(-\infty , 8]$

[-8, 8]

Correct answer:

$[8, \infty )$

Explanation:

The domain of the sum of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

$x-8 \ge 0$

Adding 8 to both sides:

$x-8 + 8 \ge 0 + 8$

$x \ge 8$

This domain is $[8, \infty )$ .

The domain of is the set of all values of such that:

$x+8 \ge 0$

Subtracting 8 tfrom both sides:

$x+ 8-8 \ge 0 - 8$

$x \ge -8$

This domain is $[-8, \infty )$

The domain of f+g is the intersection of the domains, which is $[8, \infty )$ .

Report an Error

Example Question #41 : Algebraic Concepts

Define functions $f(x) = \sqrt{x}$ and $g(x) =- x^{2}$

Give the domain of the composition $f \circ g$ .

Possible Answers:

$\left ( 0, \infty\right )$

$\left [ 0, \infty\right )$

$\left ( - \infty, 0 \right )$

$\left ( - \infty, 0 \right ]$

None of the other choices gives the correct response.

Correct answer:

None of the other choices gives the correct response.

Explanation:

The domain of the composition of functions $f \circ g$ is the set of all values that fall in the domain of such that g(c) falls in the domain of .

$g(x) =- x^{2}$ , a polynomial function, so the domain of is equal to the set of all real numbers, $\left ( -\infty, \infty\right )$ . Therefore, places no restrictions on the domain of $f \circ g$ .

$f(x) = \sqrt{x}$ , a square root function, whose domain is the set of all values of that make the radicand nonnegative. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or $\left [ 0, \infty\right )$ . Therefore, we must select the set of all in $\left ( -\infty, \infty\right )$ such that $g(c) \ge 0$ , or, equivalently,

$-c^{2} \ge 0$

However, $c^{2} \ge 0$ , so $-c^{2} \le 0$ , with equality only if c = 0 . Therefore, the domain of $f \circ g$ is the single-element set $\left \{ 0 \right \}$ , which is not among the choices.

Report an Error

Example Question #261 : Hi Set: High School Equivalency Test: Math

Define functions $f(x) = \sqrt{x+8}$ and $g(x) = \sqrt{8-x}$ .

Give the domain of the function f-g .

Possible Answers:

$[8, \infty )$

[-8, 8]

$(-\infty , -8]$

$(-\infty , 8]$

$[-8, \infty )$

Correct answer:

[-8, 8]

Explanation:

The domain of the difference of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

$x+8 \ge 0$

Subtracting 8 tfrom both sides:

$x+ 8-8 \ge 0 - 8$

$x \ge -8$

This domain is $[-8, \infty )$

The domain of is the set of all values of such that:

$8-x \ge 0$

Adding to both sides:

$8-x + x \ge 0 + x$

$8 \ge x$ , or

$x \le 8$

This domain is $(-\infty , 8]$ .

The domain of f-g is the intersection of these, [-8, 8] .

Report an Error

Example Question #5 : Domain And Range Of A Function

Define functions $f(x) = \sqrt[3]{x-8}$ and $g(x) = \sqrt[3]{x+ 8}$ .

Give the domain of the function f+g .

Possible Answers:

$(-\infty, 8]$

$[8, \infty )$

$[-8, \infty )$

$(-\infty, - 8]$

$(-\infty, \infty)$

Correct answer:

$(-\infty, \infty)$

Explanation:

The domain of the sum of two functions is the intersections of the domains of the individual functions.

Both and are cube root functions. There is no restriction on the value of the radicand of a cube root, so both of these functions have as their domain the set of all real numbers $(-\infty, \infty)$ . It follows that f+g also has domain $(-\infty, \infty)$ .

Report an Error

Example Question #41 : Algebraic Concepts

Define

$f(x)= \left\{\begin{matrix} x-2,&x< 0 \\ x+ 3,& x \ge 0\end{matrix}\right.$

Give the range of the function.

Possible Answers:

$(-\infty, -2] \cup (3, \infty)$

(-2, 3]

$(-\infty, \infty)$

$(-\infty, -2) \cup [3, \infty)$

[-2, 3)

Correct answer:

$(-\infty, -2) \cup [3, \infty)$

Explanation:

The range of a function is the set of all possible values of f(x) over its domain.

Since this function is piecewise-defined, it is necessary to examine both parts of the function and extract the range of each.

For x< 0 , it holds that f(x) = x-2 , so

x< 0

x-2< 0-2

f(x)< -2 ,

$x \in(-\infty, -2)$ .

For $x \ge 0$ , it holds that f(x)= x+ 3 , so

$x \ge 0$

$x+ 3 \ge 0+ 3$

$f(x) \ge 3$ ,

$x \in [3, \infty)$

The overall range of is the union of these sets, or $(-\infty, -2) \cup [3, \infty)$ .

Report an Error

View Tutors

Quang
Certified Tutor

Georgetown University, Bachelor of Science, Finance.

View Tutors

Peter
Certified Tutor

Rensselaer Polytechnic Institute, Bachelor of Science, Materials Engineering. California University of Pennsylvania, Master o...

View Tutors

Metaxia
Certified Tutor

Rutgers University-Camden, Bachelor in Arts, Digital Arts. Loyola University Maryland, Masters in Education, Special Education.

All HiSET: Math Resources

125 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

LSAT Tutors in Atlanta, Chemistry Tutors in New York City, Calculus Tutors in Phoenix, SAT Tutors in Philadelphia, MCAT Tutors in Denver, MCAT Tutors in Atlanta, Calculus Tutors in Los Angeles, GRE Tutors in Dallas Fort Worth, Spanish Tutors in New York City, MCAT Tutors in Houston

Popular Courses & Classes

GRE Courses & Classes in Philadelphia, ACT Courses & Classes in New York City, GRE Courses & Classes in Denver, GRE Courses & Classes in Los Angeles, Spanish Courses & Classes in Miami, GMAT Courses & Classes in Philadelphia, GRE Courses & Classes in Miami, SAT Courses & Classes in Los Angeles, ACT Courses & Classes in Denver, ISEE Courses & Classes in Phoenix

Popular Test Prep

MCAT Test Prep in Boston, ISEE Test Prep in Philadelphia, MCAT Test Prep in Chicago, SSAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Phoenix, SAT Test Prep in Philadelphia, ACT Test Prep in Denver, SAT Test Prep in Boston, ISEE Test Prep in Denver, GRE Test Prep in Los Angeles

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

HiSET: Math : Domain and range of a function

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #1 : Domain And Range Of A Function

Example Question #2 : Domain And Range Of A Function

Example Question #5 : Functions And Function Notation

Example Question #1 : Functions And Function Notation

Example Question #1 : Functions And Function Notation

Example Question #11 : Functions And Function Notation

Example Question #41 : Algebraic Concepts

Example Question #261 : Hi Set: High School Equivalency Test: Math

Example Question #5 : Domain And Range Of A Function

Example Question #41 : Algebraic Concepts

All HiSET: Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: