Answer:
374 miles
Step-by-step explanation:
To see how many miles each gallon gets, divide 102/3 which is 34 meaning there are 34 miles per gallon so if you multiply 34 x 11 gallons it equals 374 miles.
Hope this helps!
To solve this problem, we will use the concept of a unit rate, which in this scenario is the car's miles per gallon (mpg). A unit rate compares a quantity to one unit of another quantity. Here's how to do it step by step:
1. **Find the unit rate (mpg).**
The car travels 102 miles on 3 gallons of gas. To find out how many miles it travels per gallon, we'd divide the total number of miles by the number of gallons of gas used:
\[ \text{mpg} = \frac{\text{miles traveled}}{\text{gallons used}} \]
\[ \text{mpg} = \frac{102 \text{ miles}}{3 \text{ gallons}} \]
\[ \text{mpg} = 34 \text{ miles per gallon} \]
So the car travels 34 miles per gallon of gas.
2. **Calculate the distance the car can travel on 11 gallons of gas.**
Now that we know the car travels 34 miles on one gallon of gas, we can find out how far it travels on 11 gallons by multiplying the mpg by the number of gallons:
\[ \text{distance} = \text{mpg} \times \text{gallons available} \]
\[ \text{distance} = 34 \text{ miles/gallon} \times 11 \text{ gallons} \]
\[ \text{distance} = 374 \text{ miles} \]
So, the car can travel 374 miles on 11 gallons of gas.
Solve the system using elimination:
-5x + 3y = 2 for x
-5x - 5y = -30 for y
Hey there! :)
Equation 1 : -5x + 3y = 2
Equation 2 : -5x - 5y = -30
Because both equations contain a like term (which in this case is 5x), we can easily subtract these equations from one another.
Subtract equation 1 from equation 2 :
-5x - (-5x) = 0
3y - (-5y) = 3y + 5y --> 8y
2 - (-30) = 2 + 30 --> 32
This leaves us with 8y = 32. Simplify this to get y by dividing both sides by 8.
8y ÷ 8 = 32 ÷ 8
Simplify.
y = 4
Now, plug in 4 for y in our first equation.
-5x + 3(4) = 2
Simplify.
-5x + 12 = 2
Subtract 12 from both sides.
-5x = 2 - 12
Simplify.
-5x = -10
Divide both sides by -5.
-5x ÷ -5 = -10 ÷ -5
Simplify.
x = 2
Therefore, our answer is :
(2, 4) where x = 2, y = 4
~Hope I helped!~
If cos = 3/5, then tan = _____.
cramming trying to graduate please help
Answer:4/3
Step-by-step explanation:BY PHYTHAGORAS THEOREM
OPP=X,ADJ=3,HYP=5
5^2=3^2+X^2
25=9=X^2
25-9=X^2
16=X^2
4=X
BY TRIGONOMERTRY
SOHCAHTOA
TAN=OPP/ADJ
=4/3
The answer is:
[tex]Tan(\beta)=Tan(53.13)=1.33\°[/tex]
or
[tex]Tan(\beta)=\frac{4}{3}[/tex]
Why?To find the answer, we first need to find the value of the given angle, we can calculate it since we already know the cosine of that angle.
So, solving we have:
[tex]Cosine(\beta)=\frac{3}{5} \\\\Cosine((\beta)^{-1}=Arccos(\frac{3}{5})\\\\\beta =53.13\°[/tex]
Now, that we know that the angle is equal to 53.13°, we can calculate the value of the tangent, so:
[tex]Tan(\beta)=Tan(53.13)=1.33\°[/tex]
Also, we can find it using the Pythagorean Theorem and the trigonometric identities:
We have that:
[tex]Cosine=\frac{Adjacent}{Hypothenuse}[/tex]
So, using the given information we have:
[tex]Cosine=\frac{3}{5}[/tex]
Where,
Hypothenuse, is equal to 5.
Adjacent, side is equal to 3.
So, we are looking for the opposite side.
Then, substituting it into the Pythagorean Theorem formula, we have:
[tex]Hypothenuse^{2}=Adjacent^{2}+Opposite^{2}\\\\5^{2}=3^{2}+Opposite^{2}\\\\Opposite^{2}=25-9=16\\\\Opposite=\sqrt{16}=4[/tex]
Now that we already know the opposite side, we can find the value of the tangent that will be:
[tex]Tan(\beta)=\frac{Opposite}{Adjacent}=\frac{4}{3}[/tex]
Have a nice day!
if 2k, 5k-1 and 6k+2 are the first 3 terms of an arithmetic sequence, find k and the 8th term
Answer:
see explanation
Step-by-step explanation:
The common difference d of an arithmetic sequence is
d = [tex]a_{2}[/tex] - [tex]a_{1}[/tex] = [tex]a_{3}[/tex] - [tex]a_{2}[/tex]
Substitute in values and solve for k, that is
5k - 1 - 2k = 6k + 2 - (5k - 1)
3k - 1 = 6k + 2 - 5k + 1
3k - 1 = k + 3 ( subtract k from both sides )
2k - 1 = 3 ( add 1 to both sides )
2k = 4 ⇒ k = 2
--------------------------------------------------------
The n th term of an arithmetic sequence is
[tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n - 1)d
[tex]a_{1}[/tex] = 2k = 2 × 2 = 4 and
d = 5k - 1 - 2k = 3k - 1 = (3 × 2) - 1 = 5
Hence
[tex]a_{8}[/tex] = 4 + (7 × 5) = 4 + 35 = 39
what is the distance between the points (-6, 7) and (-1, 1)
Answer:
sqrt(61)
Step-by-step explanation:
The distance between two points is found by the following formula
d = sqrt (( x2-x1)^2 + (y2-y1)^2 )
= sqrt( (-1--6)^2 + (1-7)^2)
= sqrt( (-1+6)^2 + (1-7)^2)
= sqrt( (5)^2 + (6)^2)
= sqrt( 25 + 36)
= sqrt(61)
Write the slope-intercept form of the equation for the line.
Answer:
d
Step-by-step explanation:
What is the degree measure of r?
A) 27°
B) 32°
C) 37°
D) 64°
Answer:
r=64°
Step-by-step explanation:
angles in a triangle = 180°
180 -37 -27 = 116
angles in a straight line = 180°
180 -116 = 64
r= 64
Answer:
d. 64
Step-by-step explanation: i had it on usa testprep
Which BEST describes the syntax in this excerpt?
A) The arrangement of words and phrases indicates erroneous judgment and bias.
B) The arrangement of words and phrases suggests repressed childhood memories.
C) The arrangement of words and phrases implies affection and compassion for family.
D) The arrangement of words and phrases suggests a thought process that flits from thought to thought.
Answer:
The answer is D.The arrangement of words and phrases suggests a thought process that flits from thought to thought.
Step-by-step explanation:
The best description of the syntax in the given text indicates that the arrangement of words and phrases suggests a thought process that moves erratically from one thought to another, reflecting the text's stream-of-consciousness style and mixed sentence constructions.
Explanation:The question involves analyzing an excerpt for the syntax used, which refers to the manner by which words are organized into sentences. Among the given choices, the best description that matches the details provided about the text would be D) The arrangement of words and phrases suggests a thought process that flits from thought to thought. This is supported by the description of the text which includes mixed sentence constructions, lack of consistency in grammatical paths, and an attempt by the author to meet or challenge conventional expectations in rhetorically effective ways. The text also mentions the replication of a child's growing awareness and a stream-of-consciousness style which implies a thought process that is not linear but more erratic and intuitive, which is characteristic of syntax that flits from thought to thought.
if a square has a side of 12xy+1, it’s perimeter is:
Answer:
48xy+4
Step-by-step explanation:
Because we know that a square has 4 sides, we can multiply this expression by 4, thus ending in 48xy+4. Hope this helps :)
For this case we have by definition, that the perimeter of a square is given by the sum of its sides. As the sides measure the same, then the perimeter is:
[tex]p = 4l[/tex]
Where "l" represents the side of the square:
Then, they tell us that:
[tex]l = 12xy + 1[/tex]
So:
[tex]p = 4 (12xy + 1)\\p = 48xy + 4[/tex]
ANswer:
[tex]p = 48xy + 4[/tex]
PLEASE HELP
Match the pairs of equivalent expressions
[tex](4t-\dfrac{8}{5})-(3-\dfrac{4}{3}t)---------------\dfrac{16t}{3}-\dfrac{23}{5}[/tex]
[tex]5(2t+1)+(-7t+28)--------------3t+33[/tex]
[tex](\dfrac{-9}{2}t+3)+(\dfrac{7}{4}t+33)-----------------\dfrac{-11t}{4}+36[/tex]
[tex]3(3t-4)-(2t+10)-------------7t-22[/tex]
Step-by-step explanation:a)
[tex](4t-\dfrac{8}{5})-(3-\dfrac{4}{3}t)[/tex]
Firstly we will open the parentheses term and then on combining the like terms and then solving them.
[tex]=4t-\dfrac{8}{5}-3+\dfrac{4}{3}t\\\\i.e.\\\\=4t+\dfrac{4}{3}t-\dfrac{8}{5}-3\\\\=\dfrac{4t\times 3+4t}{3}+\dfrac{-8-3\times 5}{5}\\\\=\dfrac{16t}{3}-\dfrac{23}{5}[/tex]
b)
[tex]5(2t+1)+(-7t+28)\\\\=5\times 2t+5\times 1-7t+28\\\\=10t+5-7t+28\\\\=10t-7t+5+28\\\\=3t+33[/tex]
( Here we used distributive property for first bracket and open the parentheses term and them we combined the like terms and finally simplified our terms )
c)
[tex](\dfrac{-9}{2}t+3)+(\dfrac{7}{4}t+33)[/tex]
We firstly open our parentheses term but there will be no change in the terms since the sign before parentheses is positive.
and then we will combine the like terms and simplify them.
[tex]=\dfrac{-9}{2}t+3+\dfrac{7}{4}t+33\\\\\\=\dfrac{-9}{2}t+\dfrac{7}{4}t+3+33\\\\\\=\dfrac{-9t\times 2+7t}{4}+36\\\\\\=\dfrac{-11t}{4}+36[/tex]
d)
[tex]3(3t-4)-(2t+10)[/tex]
i.e. we use the distributive property for first bracket and open second parentheses term. since the sign before the second parentheses is negative hence the sign of the terms will get interchanged.
i.e.
[tex]=3\times 3t+3\times (-4)-2t-10\\\\i.e.\\\\=9t-12-2t-10\\\\=9t-2t-12-10\\\\=7t-22[/tex]
which statement about the function Y equals X2 +5 is correct
Answer: what are the options?
Step-by-step explanation:
Plz help me with this
Answer: B) y = -4 sin(x - π/2)
Step-by-step explanation:
The graph of a sine function is the cosine function shifted π/2 units to the right.
Similarly, the graph of a cosine function is the sine function shifted π/2 units to the left.
(4x^2+3x+7)(8x-5) multiply the polynomials
Answer:
[tex]32x^3+4x^2+41x-35[/tex]
Step-by-step explanation:
We have these two polynomials
[tex](8x-5)(4x^2+3x+7)[/tex]
Now we need to distribute one polynomial to all of the terms to the other.
[tex]32x^3+24x^2+56x-20x^2-15x-35[/tex]
Next we can combine all of the like terms
[tex]32x^3+24x^2+56x-20x^2-15x-35\\\\32x^3+4x^2+41x-35[/tex]
Answer: 32x^3 + 4x^2 + 41x - 35
Step-by-step explanation:
Find the solution set of this inequality |11x - 22| > 22
Need help ASAP!!
Your answer should be x < 0 or x > 4.
I'm 3 Brainliests away from ranking up, so one would be much appreciated. Thank you, and good luck!
If B =90°,ac=96cm,c=30° what dose ab = in cm
Answer: 48
96/2 = 48 cm
Answer:
ab = 48 cm
Step-by-step explanation:
Supposing ABC is a right angled triangle where m∠B = 90°, ac = 96 cm and c = 30°, we are to find the length of the side ab in cm.
For this, we will use the trignometric ratios to find ab.
[tex] sin B = \frac { a b } { a c } [/tex]
[tex] sin 30 = \frac { a b } { 9 6 } [/tex]
[tex]\frac{1}{2} =\frac{ab}{96}[/tex]
[tex]96=2ab[/tex]
[tex]ab=\frac{96}{2}[/tex]
ab = 48 cm
Volume of prisms just 3 problems
Answer:
2. 18 cubic cm.
3. 27 cubic cm
5. 90 cubic m
Step-by-step explanation:
2. Triangular prism
So, we have a triangular prism with a width of 3 cm, a height of 1.5 cm and a length of 4. We'll take the triangle shape as the base.
Like for any prism, the volume is the base multiplied by the height.
So, the area of the base is found with the regular formula: base x height:
so, 3 x 1.5 = 4.5 sq cm for the base.
Then we multiply the base by the length to get the volume: 4.5 x 4 = 18 cubic cm.
3. Trapezoidal prism
Again, first step is to calculate the area of the trapezoid shape, then multiply by its depth.
The area of a trapezoid is calculated with the formula:
A = h/2 * (a + b) where a and b represent the short and long sides respectively.
So, for our given trapezoid:
A = 1.5/2 * (2+4) = 0.75 * 6 = 4.5 sq cm.
Then we multiply the base area by the height to get the volume:
V = 4.5 * 6 = 27 cubic cm
5. Triangular prism 2
The best way to approach this one is to consider it's a triangle prism with a height of 3 cm, and use the triangle shape (with sides of 13 and 5) as the base.
But to calculate the area of that triangle we need to find out the missing length. Using Pythagore's theorem, we can easily find the length of the other side with the hypotenuse formula:
hypotenuse² = sideA² + sideB²
In our case, we have:
13² = x² + 5²
169 = x² + 25
x² = 144
x = 12
Now we can calculate the area of the triangle easily:
A = (b x h)/2 = (5 x 12)/2 = 30 sq m
Then multiply by the thickness of the prism....
V = 30 x 3 = 90 cubic m
Determine the axis of symmetry for the function f(x)= -2(x + 3)2 - 5.
Answer:-2.125
Step-by-step explanation:
1. simplify the function
-2(x+3)2-5
(-2x-6)2-5
(-4x-12)-5
-4x-17
2. formula for axis of symmetry: -b/2a
a=-4 b=-17
-(-17)/2(-4)=17/-8=-2.125
Answer:
The axis of symmetry for the function f(x) is x=-3.
Step-by-step explanation:
The vertex form of a quadratic function is
[tex]y=a(x-h)^2+k[/tex] .... (1)
Where, (h,k) is vertex and x=h is the axis of symmetry.
The given function is
[tex]f(x)=-2(x+3)^2-5[/tex] .... (2)
From (1) and (2), we get
[tex]a=-2,h=-3,k=-5[/tex]
The axis of symmetry is
[tex]x=h[/tex]
Substitute h=-3 in the above equation.
[tex]x=-3[/tex]
Therefore the axis of symmetry for the function f(x) is x=-3.
5) In -x + In 8 = In 36
Properties of logarithms
Answer:
x = -4.5
Step-by-step explanation:
We shall be using the following property of logarithms;
[tex]lna+lnb=ln(a*b)[/tex]
Therefore, In -x + In 8 can be re-written as;
[tex]ln-x+ln8=ln(-8x)\\\\ln(-8x)=ln36\\\\-8x=36\\\\x=-4.5[/tex]
a tree casts a 20-foot Shadow at the same time the person who is 6 feet tall casts an shadow 18 ft Shadow how big is the tree
6 ft person casts 18 ft shadow
x ft tree casts 20 ft shadow
6*20 = 18x
120 =18x
x=120/18
×≈6.67
The height of the tree is 6.67 feet.
What is proportionality?The term proportionality describes any relationship that is always in the same ratio.
Given that, a tree casts a 20-foot shadow at the same time the person who is 6 feet tall casts a shadow 18 ft, we need to find the height of the tree.
Here, using the concept of proportionality, since, the phenomena is happening at the same time, therefore, the ratio of the height and length if the shadow will be equal for all.
Let the height of the tree be x,
20 / x = 18 / 6
120 = 18x
x = 6.67
Hence, the height of the tree is 6.67 feet.
Learn more about proportionality, click;
https://brainly.com/question/29126727
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What is the range of the absolute value function below?
answer: f(x)<4
I really hope that heped
Answer:
D The bottom choice.
Step-by-step explanation:
The range is the y value of (in this case) an absolute value. The maximum value of this is 4. Everything else is less than 4. The answer would be written as
4≥f(x) which is the same thing as D
Help me plsss ill give ten pointsssss
Answer:
A
Step-by-step explanation:
AE=EC
5x-10=2x+5
5x-2x=5+10
3x=15
x=5
Answer:
The correct answer is option A. 5
Step-by-step explanation:
It is given a rectangle ABCD
To find the value of x
It is given that,
AE = 5x - 10 and EC = 2x + 5
Since ABCD is a rectangle, All diagonals are equal.
AC = BD and AE = EC
AE = EC
5x - 10 = 2x + 5
5x - 2x = 5 + 10
3x = 15
x = 15/3 = 5
Therefore the correct answer is option A. 5
Rick is taking a 1-mile ride on his bicycle. For each rotation of his tire, he travels 72 inches. How many rotations will his tire make during the 1-mile trip?
73 rot
Answer:
880
Step-by-step explanation:
First we have to convert 1 mile into inches.
1 mi * 5280 ft/mi * 12 in/ft = 63,360 inches
Each rotation covers 72 inches, so the number of rotations is:
63,360 in / (72 in / rotation) = 880 rotations
Answer:
880
Step-by-step explanation:
I got it right on my test.
The vertices of a quadrilateral are A(-1,6),B(-2,4),C(2,2)and D(3,4) write a paragraph proof to determine whether quadrilateral abcd is a rectangle
Answer:
The given quadrilateral is a rectangle.
Step-by-step explanation:
We are given the vertices of a quadrilateral,
We can use the slopes and modulus of all sides to determine whether it is a rectangle or not. The four sides of the quadrilateral are Ab, BC, CD and AD. So finding slopes of all sides using the formula
[tex]Slope=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
where x_1 and y_1 are the coordinates of firsst vertex and x_2 and y_2 are coordinates of second vertex.
So.
Slopes are:
Slope of AB = 2 and Length=2.23
Slope of BC = -1/2 and length = 4.47
Slope of CD = 2 and length = 2.23
Slope of AD = -1/2 and length = 4.47
As we can see that the slope of AB and CD is equal which means that they are parallel and their lengths are also same.
Similarly BC and AD have same slope which means that they are also parallel and their lengths are also equal.
More over the product of slopes of AB and CD, BC and CD, CD and AD and AD and AB is equal to -1 which indicated that these sides are parallel to each other so it can be concluded that as the opposite sides of the quadrilateral are equal and the interior angles are 90 degrees so the quadrilateral is a rectangle ..
Answer:
I am given that the vertices of quadrilateral ABCD are A(-1, 6), B(-2, 4), C(2, 2), and D(3, 4). The slope formula applied to each pair of adjacent vertices gives the slopes (m) of the sides:
slope of AB (m AB) = 4 - 6 / -2 - -1) = -2 / -1 =2
slope of BC (m BC) = 2 - 4 / 2 - (-2) = -2 / 4 = - 1 / 2
slope of CD (m CD) = 4 - 2 / 3 -2 = 2 / 1 = 2
slope of DA (m DA) = 6 - 4 / - 1 -3 = 2 / - 4 = - 1 / 2
Because line segments with equal slopes are parallel, segment AB is parallel to segment CD and segment BC is parallel to segment DA.
Multiplying the slopes of one pair of adjacent sides, I find that.
mAB * mBC = 2 * - 1/2 = - 1
If the product of the slopes of two segments is -1, then they are perpendicular. Since both pairs of opposite sides have been proven parallel, proving one pair of adjacent sides perpendicular implies that the other pair of adjacent sides are also perpendicular. Therefore, by definition, quadrilateral ABCD is a rectangle.
Step-by-step explanation:
5. Suppose the stacks of cases of bottled water on the shelves of a major grocery store follow
the graph of the equation 4r + y= 36 where r is hours after the store opens and y is the
number of cases on the shelves. All of the water is sold. What is the domain of the
variable x?
© (0,6]
® (0,91
© (0,12]
© (-2,91
ANSWER
C [0,9]
EXPLANATION
The number of bottled water on the shelf is modeled by the function
[tex]4x + y = 36[/tex]
We make y the subject to obtain;
[tex]y = 36 - 4x[/tex]
When no water is sold, then x=0.
When all the bottles are sold, then
[tex]4x - 36 = 0[/tex]
[tex]4x = 36[/tex]
[tex]x = \frac{36}{4} = 9[/tex]
Therefore the store cannot sell more than 9 bottles.
The domain is [0,9]
Find the missing side lengths
since this a 45 45 90 right triangle to get the side lengths you divide 2 times the square root of 2 and you get 2, since this has two congruent angles which makes it an Isosceles triangle. your a and b value are 2.
a water sprinkler rotates in a circular pattern witha radius of 8 feet. if the sprinkler stops 60 degrees short of completing the circular patterns, which is closest to the distance of the outer are that is not getting sprinkled with water
a.4.19 ft
b.8.37ft
c.41.87ft
d.50.24ft
Answer:
B. 8.37 ft
Step-by-step explanation:
⇒This question is on circumference
⇒The water sprinkler covers a circular pattern as it moves to cover 300° and remain with 60° to cover so as to complete a single circular pattern.
⇒The remaining distance asked is the one represented by 60°
⇒Apply the formula for circumference of a circle with angle theta
[tex]c=2\pi r\alpha/360[/tex] where alpha is the angle in degrees
Given that r=8ft and ∝=60°
c=3.14×2×8×60°/360°
c=8.37 ft
What equation represent the relationship between a and b show in the table ?
[tex]b=a+5[/tex]
Hope this helps.
Answer:
b=a+5
Step-by-step explanation:
6-1=5
7-2=5
8-3=5
9-4=5
Please rate brainliest! :D
PLEASE HELP! URGENT! WHAT'S "Y"
Answer:
y-axis The vertical number line in a coordinate graph. The line in the coordinate plane, usually vertical, or in space, containing those points whose first coordinates (and third, in space) are 0. y-coordinate The second coordinate of an ordered pair or ordered triple.
Step-by-step explanation:
which function has the same y-intercept as the function y equals 2/3 x - 3
We have the function [tex]y = \frac{2}{3} x -3[/tex] and we want to find a function that has the same y-intercept than the previous function.
First, let's find the y-intercept by subtituting 0 for 'x'.
[tex]y = \frac{2}{3} (0) -3 = -3[/tex]
Now that we found that y-intercept =-3, any lineal function of the type: [tex]y = ax - 3[/tex] will have the same y-intercept. Where 'a' can take all the real values.
Also, any quadratic function of the type: [tex]y=ax^{2} + bx - 3[/tex] will have the same y-intercept. Where 'a' and 'b' can take all the real values.
Hey guys
I needed a help
Solve 2x -3/x=1
Answer:
see explanation
Step-by-step explanation:
Given
2x - [tex]\frac{3}{x}[/tex] = 1
Multiply through by x
2x² - 3 = x ( subtract x from both sides )
2x² - x - 3 = 0 ← in standard form
To factorise the quadratic
Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term.
The factors are + 2 and - 3
product = 2 × - 3 = - 6 and sum = 2 - 3 = - 1
Use these factors to split the x- term
2x² + 2x - 3x - 3 = 0 ( factor the first/second and third/fourth terms )
2x(x + 1) - 3(x + 1) = 0 ← factor out (x + 1) from each term
(x + 1)(2x - 3) = 0
Equate each factor to zero and solve for x
x + 1 = 0 ⇒ x = - 1
2x - 3 = 0 ⇒ 2x = 3 ⇒ x = [tex]\frac{3}{2}[/tex]
Which of the following is the best definition of the domain
Answer:
Step-by-step explanation:
It seems that the question was cut off before it was completed. To provide an answer related to the domain in mathematical terms, the domain is an essential component of a function. Let me explain it in detail:
In mathematics, when we talk about a function, f(x), the domain of the function represents the set of all possible input values (x-values) for which the function is defined. It's essentially the set of "allowable" values that we can plug into our function to get an output.
To determine the domain of a given function, we have to consider the nature of the function itself and any mathematical rules or constraints that apply. Here are several examples that can come up:
1. **For a polynomial function**, such as f(x) = x^2 - 3x + 2, there's no restriction on the x-values. So the domain is all real numbers, often written as (-∞, ∞).
2. **For a rational function** (a ratio of two polynomials), like f(x) = (2x + 1) / (x - 3), we must exclude x-values that would make the denominator equal to zero, since division by zero is undefined. In this case, the domain is all real numbers except x = 3, or in interval notation: (-∞, 3) U (3, ∞).
3. **For a square root function**, such as f(x) = √(x - 4), the expression inside the square root must be non-negative, since we're typically considering real-valued functions. So the domain of this function is x ≥ 4, or in interval notation, [4, ∞).
4. **For a logarithmic function**, like f(x) = log(x), the argument of the logarithm must be positive. Therefore, the domain is x > 0, or in interval notation, (0, ∞).
5. **For trigonometric functions**, such as f(x) = sin(x), since there are no restrictions on the input of sine function in terms of real numbers, the domain is all real numbers.
In conclusion, to define the domain of a particular function, we need to consider the mathematical constraints such as not dividing by zero, keeping the radicand of square roots non-negative, and ensuring the arguments of logarithms positive, among others. Once these conditions are taken into account, we can specify the domain of the function, which is often expressed as an interval or a set of intervals on the real number line.