Q&A
Can a function’s domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding domains and ranges of the toolkit functions
We will now return to our set of toolkit functions to determine the domain and range of each.

For the
constant function
$\text{\hspace{0.17em}}f(x)=c,\text{\hspace{0.17em}}$ the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant
$\text{\hspace{0.17em}}c,\text{\hspace{0.17em}}$ so the range is the set
$\text{\hspace{0.17em}}\left\{c\right\}\text{\hspace{0.17em}}$ that contains this single element. In interval notation, this is written as
$\text{\hspace{0.17em}}[c,c],\text{\hspace{0.17em}}$ the interval that both begins and ends with
$\text{\hspace{0.17em}}c.$
For the
identity function
$\text{\hspace{0.17em}}f(x)=x,\text{\hspace{0.17em}}$ there is no restriction on
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Both the domain and range are the set of all real numbers.
For the
absolute value function
$\text{\hspace{0.17em}}f(x)=\left|x\right|,\text{\hspace{0.17em}}$ there is no restriction on
$\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.
For the
quadratic function
$\text{\hspace{0.17em}}f(x)={x}^{2},\text{\hspace{0.17em}}$ the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.
For the
cubic function
$\text{\hspace{0.17em}}f(x)={x}^{3},\text{\hspace{0.17em}}$ the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.
For the
reciprocal function
$\text{\hspace{0.17em}}f(x)=\frac{1}{x},\text{\hspace{0.17em}}$ we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write
$\left\{x\right|\text{}x\ne 0\},$ the set of all real numbers that are not zero.
For the
reciprocal squared function
$\text{\hspace{0.17em}}f(x)=\frac{1}{{x}^{2}},$ we cannot divide by
$0,$ so we must exclude
$0$ from the domain. There is also no
$x$ that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.
For the
square root function
$\text{\hspace{0.17em}}f(x)=\sqrt[]{x},\text{\hspace{0.17em}}$ we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is defined to be positive, even though the square of the negative number
$\text{\hspace{0.17em}}-\sqrt{x}\text{\hspace{0.17em}}$ also gives us
$\text{\hspace{0.17em}}x.$
For the
cube root function
$\text{\hspace{0.17em}}f(x)=\sqrt[3]{x},\text{\hspace{0.17em}}$ the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).
How To
Given the formula for a function, determine the domain and range.

Exclude from the domain any input values that result in division by zero.
Exclude from the domain any input values that have nonreal (or undefined) number outputs.
Use the valid input values to determine the range of the output values.
Look at the function graph and table values to confirm the actual function behavior.
Questions & Answers
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)

how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching

bsc F. y algebra and trigonometry pepper 2

given that x= 3/5 find sin 3x

remove any signs and collect terms of -2(8a-3b-c)

is that a real answer

Joeval

x²-2x+9-4x²+12x-20
-3x²+10x+11

Miranda

x²-2x+9-4x²+12x-20
-3x²+10x+11

Miranda

(X2-2X+8)-4(X2-3X+5)=0 ?

master

The anwser is imaginary
number if you want to know The anwser of the expression
you must arrange The expression and use quadratic formula To find the
answer

master

The anwser is imaginary
number if you want to know The anwser of the expression
you must arrange The expression and use quadratic formula To find the
answer

master

X2-2X+8-4X2+12X-20=0
(X2-4X2)+(-2X+12X)+(-20+8)= 0
-3X2+10X-12=0
3X2-10X+12=0
Use quadratic formula To find the answer
answer (5±Root11i)/3

master

Soo sorry (5±Root11* i)/3

master

x2-2x+8-4x2+12x-20
x2-4x2-2x+12x+8-20
-3x2+10x-12
now you can find the answer using quadratic

Mukhtar

explain and give four example of hyperbolic function

What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?

Find nth derivative of eax sin (bx + c).

Find area common to the parabola y2 = 4ax and
x2 = 4ay.

Anurag

y2=4ax= y=4ax/2. y=2ax

akash

A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden

to find the length I divide the area by the wide wich means 1125ft/25ft=45

Miranda

What do you call a relation where each element in the domain is related to only one value in the range by some rules?

given 4cot thither +3=0and 0°<thither <180°
use a sketch to determine the value of the following
a)cos thither

nothing up todat yet

Miranda

aap konsi country se ho

jai

which language is that

Miranda

what is the formula for calculating algebraic

I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it

Miranda

state and prove Cayley hamilton therom

the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.

Miranda

Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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