The parametric equations that represent the sphere surface between the planes z=±6 are x = 12*sin(θ)*cos(φ), y = 12*sin(θ)*sin(φ), and z = 12*cos(θ), with θ and φ being between π/3 and 2π/3, and 0 and 2π respectively.
Explanation:This question involves a mathematical concept known as parametric representation of a surface, specializing in spheres and planes. In a 3D coordinate system, the equation of a sphere is given by x² + y² + z² = r² where r is the radius of the sphere, but we restrict the z variable to lie in a certain range, which is between -6 and 6 in this situation.
To represent the sphere in parametric form, we often use spherical coordinates. The relationships between Cartesian coordinates and spherical coordinates are as follows: x = r*sin(θ)*cos(φ), y = r*sin(θ)*sin(φ), z = r*cos(θ). Here r = √144 =12, is the radius of the sphere.
Given the restrictions z = ±6, we have cos(θ) = ±6/12 = ±1/2, or θ = π/3 and 2π/3. So, the spherical coordinates range within 0 <= φ <= 2π and π/3 <= θ <= 2π/3.
This gives the parameters of the restricted surface in terms of θ and φ those are as follows: x = 12*sin(θ)*cos(φ), y = 12*sin(θ)*sin(φ), and z = 12*cos(θ).
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The parametric equations which represent the part of the sphere x² + y² + z² = 144 that lies between the planes z = -6 and z = 6 are: x = 12*sin(θ)*cos(Ď•), y = 12*sin(θ)*sin(Ď•), z = 12*cos(θ), with 60° ≤ θ ≤120° and 0° ≤ Ď• ≤ 360°.
Explanation:The question asks for a parametric representation of part of a sphere lying between two planes. The given sphere equation is x² + y² + z² = 144, and the two planes are defined by z = -6 and z = 6. This is a common problem in multivariable calculus, and we solve it using spherical coordinates. In spherical coordinates, x= r*sin(θ)*cos(Ď•), y= r*sin(θ)*sin(Ď•) and z= r*cos(θ) where r is the radius, θ is the inclination angle, and Ď• is the azimuthal angle. For the given sphere, r = √144 = 12.
The bounds for z adds constraints to our inclination angle θ. Given -6 ≤ z ≤ 6, we determine corresponding bounds on θ. For z = r*cos(θ), we have -6 ≤ 12cos(θ) ≤ 6, or -1/2 ≤ cos(θ) ≤ 1/2. This yields θ values between 60° and 120°. The parametric equations describing the part of the sphere between the planes z = -6 and z = 6 are:
x = 12*sin(θ)*cos(Ď•)
y = 12*sin(θ)*sin(Ď•)
z = 12*cos(θ)
where 60° ≤ θ ≤120° and 0° ≤ Ď• ≤ 360°.
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Algebra II Please Help
A new mountain bike is on sale for 260.00, which is 35% off of is original price. What is the original price of the bike
Answer:g hvt
Step-by-step explanation:
please help will give brainliest
What is the period of the function f(x) shown in the graph?
Answer:
The period is [tex]\pi[/tex].
Step-by-step explanation:
The answer is determined by finding the length of 1 cycle of the function. Trig functions sine and cosine have a wavy form. One starts and ends int he middle - sine. The other starts above and stops above- cosine. Regardless of the function, the period will be the same because the period asks "how fast does this function complete 1 wave?"
If cosine we start at one of the top points, trace the graph till we reach the same y-value again. The period is the time or difference in x-values. For instance if I started at the x-value [tex]\frac{\pi }{2}[/tex] and ended at [tex]\frac{3\pi }{2}[/tex] then I'd subtract them.
[tex]\frac{3\pi }{2}[/tex]-[tex]\frac{\pi }{2}[/tex]=[tex]\frac{2\pi }{2}[/tex]=[tex]\pi[/tex]
The period is [tex]\pi[/tex].
What is the domain and range of the function
Answer:
Option d is correct.
Domain = all real number
range = positive real numbers
Step-by-step explanation:
Given the function: [tex]y=f(x) = a^x[/tex]
Domain of the function is all real numbers except where the expression is undefined.
In this case, there is no real number that makes the expression undefined.
Domain of f(x) = [tex](-\infty , \infty)[/tex] = {a | a∈R}
Range is the set of all valid f(x) values.
[tex](0,1) \cup (1, \infty)[/tex]
[tex]\{y | y\neq 1 , y>0\}[/tex]
Therefore, Domain is all real number and the range of function is positive real number
help me
important
thanks ppl
Answer:
A
Step-by-step explanation:
Your first step is to figure out what f(x) is. You should start with x = 0. f(x) has a value of - 4. That only tells you that whatever x is or what it is coupled with, the constant term is - 4.
What you know so far is that y = x something - 4
Now go to work on some of the other numbers. It really doesn't look linear so don't try it.
What happens when x = 16? Somehow f(x) winds up being 0. What can cause that?
First of all it can't be y = ax - 4. We've already establish the constant must be - 4
What about y = ax^2 - 4
That cannot be either. y = 16^2*a - 4 ?? y = 256ax - 4. No other value will fit the bill.
By looking at g(x) you get the idea that y = a*sqrt(x) - 4 might have something to do with f(x). The most obvious value for a = 1, so try it first.
y = sqrt(x) - 4y = sqrt(16) - 4y = 4 - 4 = 0Looking good!!Now try some of the other values.
x = 9y = sqrt(9) - 4y = 3 - 4y = -1And that's what the table says.
The red graph is y = sqrt(x) - 4
The blue graph is y = 4*sqrt(x) - 8
Answer
It looks like A is the answer.
g(x) has a y intercept of - 8
f(x) has a y intercept of - 4
==============
The two intercepts are not equal. If they were they would start from the same point. B is incorrect
==============
C is wrong. the x intercepts are distinct points. One is 4 and the other is 16 I think.
0 = sqrt(x) - 4
4 = sqrt(x) Square both sides.
16 = x This is the x intercept of the red line
The other intercept = 4 (Blue line). I'll leave you to work it out. Leave a not if you can't.
========================
A and D can't both be true. Since A is true, D can't be.
Determine the maximum numbers of zeros of the polynomial function 3x^4-x^2+1
A. 4
B. 3
C. 1
D. 2
Answer: A. 4
The largest exponent in the polynomial tells us the max number of roots, x intercepts, or zeroes of the function. In this case, that happens to be 4. This is the degree of the polynomial. It is considered a quartic polynomial. It is also a trinomial since it has 3 terms (3x^4, x^2 and 1)
Answer:
ITs A
Step-by-step explanation:
The highest degree is 4 ( 3x^4) so the maximum number of zeroes is 4.
In quadrilateral LMNO, LO ∥ MN. What additional information would be sufficient, along with the given, to conclude that LMNO is a parallelogram? Check all that apply. ML ∥ NO ML ⊥ LO LO ≅ MN ML ≅ LO MN ⊥ NO
Answer:
In quadrilateral LMNO, LO ∥ MN.
ML ∥ NO
LO ≅ MN
Answer:
1 & 3 is correct on edg.
A six sided number cube is rolled 12 times. An even number is rolled 7 times. What is the experimental probability that an even number is rolled?
Answer:
The experimental probability is 7/12 or 58.3%
Step-by-step explanation:
In order to find the experimental probability, we just look at the amount of times it happened in our test. The fact that the actual probability is 50% does not matter when looking at experimental numbers.
math help will be marking brainliest
Juan runs an experiment. He flips a coin 20 times. It landed on Heads 13 times and Tails 7 times. Find the experimental probability.
Compare your experimental probability to the theoretical probability of flipping a coin and it landing on heads.
Which of the following statements is true?
Question 5 options:
The theoretical probability is greater than the experimental probability.
The experimental probability is greater than the theoretical probability.
The experimental probability is equal to the theoretical probability.
Can not determine which probability is greater.
C it the answer because since heads hit 14 times it haves a higher percent of hitting heads again.
Write an equivalent expression to 2x + 3 + 5x + 6 by combining like terms then use x=3 to show expressions are equivalent
Answer:
7x + 9
Yes they are equivalent
Step-by-step explanation:
The first expression given to us is
2x + 3 + 5x + 6
Let us call it equation (i)
Now we have to find equivalent expression to it
the given equation is 2x + 3 + 5x +6
similar terms are terms involving x and not involving x
so solving the equation gives
2x + 5x + 3 +6
= 7x + 9
Now we have two equations one is
2x + 3 +5x + 6 ...............(i)
Other is
7x + 9 ..............(ii)
To check they are equivalent or not
Put x = 3 in equation (i)
2(3)+3+5(3)+6= 6 + 3 + 15 + 6
=30
Now put x=3 in second equation
7(3)+9=21 + 9
=30
As the answer of both the equations are 30 so they are equivalent
To write an equivalent expression to 2x + 3 + 5x + 6 by combining like terms, the expression can be simplified to 7x + 9. Substituting x with 3 in both expressions shows that they have the same value, which is 30.
Explanation:To write an equivalent expression to 2x + 3 + 5x + 6 by combining like terms, we add the coefficients of the like terms. In this case, the like terms are the terms with the same variable, which is x. So, 2x + 3 + 5x + 6 simplifies to 7x + 9. To show that this expression is equivalent to the original expression, we can substitute x with 3. So, when we replace x with 3, both expressions 2x + 3 + 5x + 6 and 7x + 9 result in the same value, which is 30.
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Please help ASAP!!!
triangle a'b'c' is a dilation of triangle abc. which is the correct description of the dilation?
Answer: A
Step-by-step explanation:
[tex]\overline{AB}[/tex] has a length of 6
[tex]\overline{A'B'}[/tex] has a length of 6 + 12 = 18, which is 3 times the length of [tex]\overline{AB}[/tex]
B = B' so this will be the center point
⇒ center B and a scale factor of 3
Answer:
Option A is the answer.
Step-by-step explanation:
Triangle A'B'C' is a dilation of triangle ABC.
We can apply the formula to get the scale factor as
Scale factor = [tex]\frac{\text{One side of triangle A'B'C'}}{\text{Corresponding side of triangle ABC}}[/tex]
= [tex]\frac{A'B'}{AB}[/tex]
= [tex]\frac{AB'+AA'}{AB}[/tex]
= [tex]\frac{6+12}{6}[/tex]
= [tex]\frac{18}{6}[/tex]
= 3
Therefore, ΔABC has been dilated with a scale factor of 3 about the point B as the center.
Option A is the answer.
If y varies directly with x and y = 2 when x = 10, then what is the value of y when x = 40? A. 8 B. 45 C. 200 D. 320
Answer:
A
Step-by-step explanation:
given that y varies directly with x then the equation relating them is
y = kx ← k is the constant of variation
to find k use the given condition y = 2 when x = 10
k = [tex]\frac{y}{x}[/tex] = [tex]\frac{2}{10}[/tex] = [tex]\frac{1}{5}[/tex]
y = [tex]\frac{1}{5}[/tex] x ← equation of variation
when x = 40, then
y = [tex]\frac{1}{5}[/tex] × 40 = 8 → A
The value of y when x = 40 with the same proportion will be 8 thus option (A) is correct.
What are the ratio and proportion?The ratio is the division of the two numbers.
For example, a/b, where a will be the numerator and b will be the denominator.
Proportion is the relation of a variable with another. It could be direct or inverse.
As per the given,
If y varies directly with x.
y ∝ x
y = kx
k = y/x
Thus, k will be the same for both conditions.
2/10 = y/40
1/5 = y/40
40/5 = y
8 = y
Hence "The value of y when x = 40 with the same proportion will be 8 ".
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Several students joined a radio-controlled model club. Some had boats, some had airplanes, and some had cars. They divided into three equal groups according to which radio-controlled model each had. Another 10 students joined the radio-controlled airplane group. There are now 15 students in this group. How many students joined the radio-controlled model club in the beginning?
Answer:
15 students joined the radio-controlled model club
Step-by-step explanation:
The students joining the radio-controlled model club were divided into three equal groups i.e
a) Radio-controlled boat group
b) Radio-controlled car group
c) Radio-controlled airplane group
10 new students joined the Radio-controlled airplane group and the new sum of students in this group is 15.
This implies that there were 5 students in Radio-controlled airplane group before the joining of 10 new students.
In the starting, the three groups have equal number of students which means that Radio-controlled boat group, Radio-controlled car group, Radio-controlled airplane group have 5 students in each group. Thus, In the beginning there were total 15 students who joined the radio-controlled model club
What is the length of GH? ______cm
The union of sets P = {1, 2, 3, 4, 5, 6} Q = {4, 5, 6, 7, 8, 9} is given by which of the following?
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9}
ø
{4, 5, 6}
Answer:
The first one, because the x values are not repeated in a data set.
Find the values of m and b that make the following function differentiable.
the piecewise function f of x equals x squared when x is less than or equal to three or mx plus b when x is greater than three
For [tex]f[/tex] to be differentiable, it must be continuous, so we need to have
[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)=f(3)[/tex]
By its definition, [tex]f(3)=3^2=9[/tex]. The one-sided limits are
[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3}x^2=9[/tex]
[tex]\displaystyle\lim_{x\to3^+}f(x)=\lim_{x\to3}mx+b=3m+b[/tex]
so we require [tex]3m+b=9[/tex].
In order for [tex]f[/tex] to be differentiable at [tex]x=3[/tex], we also need to have [tex]f'(3)[/tex] exist, which requires that [tex]f'[/tex] also be continuous at [tex]x=3[/tex]. First, compute the derivatives of all pieces of [tex]f[/tex]:
[tex]f'(x)=\begin{cases}2x&\text{for }x<3\\?&\text{for }x=3\\m&\text{for }x>3\end{cases}[/tex]
[tex]f'[/tex] is continuous at [tex]x=3[/tex] if
[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)=f'(3)[/tex]
The one-side limits are
[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3}2x=6[/tex]
[tex]\displaystyle\lim_{x\to3^+}f'(x)=\lim_{x\to3}m=m[/tex]
so we need to have [tex]m=6[/tex], and moreover [tex]f[/tex] will be differentiable if we set [tex]f'(3)=6[/tex].
So with [tex]m=6[/tex], we must have [tex]3m+b=9\implies b=-9[/tex].
For the piecewise function f(x) = x^2 when x <= 3 and mx + b when x > 3, values m = 6 and b = -9 ensure differentiability at x = 3.
To make the piecewise function f(x) differentiable at x = 3, we need the two pieces, x^2 and mx + b, to smoothly connect at x = 3. This requires the values of m and b to ensure continuity of both function values and derivatives.
First, evaluate both pieces at x = 3:
x^2 when x <= 3 and mx + b when x > 3
For continuity, set these expressions equal to each other:
3^2 = m * 3 + b
This yields 9 = 3m + b. To ensure differentiability, the derivatives of both pieces must also match at x = 3:
f'(x) = 2x when x <= 3 and f'(x) = m when x > 3
For continuity of derivatives, set the derivatives equal to each other at x = 3:
2 * 3 = m
This gives m = 6. Substituting m = 6 into the continuity equation 9 = 3m + b gives b = -9.
Therefore, the values m = 6 and b = -9 make the piecewise function f(x) differentiable at x = 3.
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A. Green Grapes
B. Red Grapes
C. Red and Green Grapes cost the same
Help with these questions please!
Answer:
1. 16
2. -4
Step-by-step explanation:
First factor the expression
lim x→1 (x^3 + 5x^2 + 3x-9)/(x-1)
lim x→1 (x-1) (x+3)^2/(x-1)
Canceling the x-1 in the top and bottom
lim x→1 (x+3)^2
Let x=1
lim x→1 (1+3)^2 = 4^2 = 16
2. lim x→0 (x^2 -6x+8) /(x-2)
First factor the expression
lim x→0 (x-4) (x-2) /(x-2)
Canceling the x-2 in the top and bottom
lim x→0 (x-4)
Let x=0
lim x→0 (0-4) = -4
Filling in the table
x -.1 -.01 -.001 .001 .01 .1
f(x) -4.1 -4.01 -4.001 -3.999 -3.99 -3.9
I know this isn't helpful, but I REALLY need help on this question...
Given: f(x) = x^2 + 7x + 10. From the Rational Root Theorem, we know p=10 and q=1. List all potential roots using the ratio p/q.
Help please, and explain.
Answer: 2x + y
Step-by-step explanation:
logₐ(3) = x
logₐ(5) = y
logₐ(45) = logₐ(3²· 5)
= logₐ(3)² + logₐ(5)
= 2 logₐ(3) + logₐ(5)
= 2 x + y substituted given values (stated above)
Step-by-step explanation:
Here we make use of the laws of logarithms:
log_a(PQ) = log_a(P)+log_a(Q)
which implies the following corollary
log_a(P^2) = log_a(P)+log_a(P) = 2log_a(P)
Notice how the log of a product is reduced to the sum of the log of the factors. (Advantage is taken of this fact in the use of logarithm tables before the wide-spread use of electronic calculators (pre-70's) )
So substituting
x=log_a(3)
y=log_a(5)
we have
log_a(45) = log_a(3^2 * 5) = log_a(3^2) + log_a(5)=2log_a(3)+log_a(5)
=2x+y
FInd the slope of a line
Answer:
First I'm not exactly sure what you're asking but the equation for slope is:
Y2-Y1/X2-X1=M
Step-by-step explanation:
M is your slope
X1 and Y1 represent your first point
X2 and Y2 represent your second point
To translate you are finding the rate of change (slope) between two points
Answer: Slope = ∆y/∆x = y2-y1/x2-x1
Step-by-step explanation:
The slope of a line which is also known as the gradient can be gotten by taking the ratio of the change in coordinate of y-axis to the x-axis of the line.
Change in y axis = y2-y1
Where y2 and y1 are final length and initial length respectively on the y axis
Change in x axis = x2-x1
Where x2 and x1 are final length and initial length respectively on the x axis.
The slope of the line therefore gives;
Slope = ∆y/∆x = y2-y1/x2-x1
I really need Help Please!!!!!!!!
ABCD is a parallelogram. If side AB = 10x - 15, side BC = 5x + 7, and side CD = 6x + 9, find the value of x.
x=____
Answer:
x=6
Step-by-step explanation:
If ABCD is a parallelogram, then AB = CD
AB=CD
10x-15 = 6x+9
Subtract 6x from each side
10x-6x -15 = 6x-6x+9
4x-15 = 9
Add 15 to each side
4x-15+15 = 9+15
4x = 24
Divide by 4 on each side
4x/4 =24/4
x=6
Look at the picture.
The equation:
10x - 15 = 6x + 9 add 15 to both sides
10x = 6x + 24 subtract 6x from both sides
4x = 24 divide both sides by 4
x = 6Real estate values in a town are increasing at a rate of 14% per year. Mrs. Knoxville purchased a building for $590,000 in 2012.
How much can she expect to sell the building for in 2020, assuming this trend continues?
Enter your answer in the box. Round to the nearest whole dollar.
Answer:
$1808263
Step-by-step explanation:
We're given with the below information:-
Initial price of the building = Principal amount (P) = $590000
Rate of interest (r) = 14% = 0.14
Time in years (t) = 2020 - 2012 = 8
The formula to calculate the final amount where interest is continuously compounded each year is :-
[tex]A = Pe^{rt}[/tex] [e is the mathematical const = 2.71828]
Plugging in the values of P, e, r, and t in the above formula, we get
[tex]A = 590000*2.71828^{0.14*8}[/tex]
=> A= 1808262.6
=> A = $1808263 (rounded off to the nearest whole dollar)
So, she expect to sell the building in 2020 for an amount of $1808263
Answer:\
180000
Step-by-step explanatii took
Nick currently has 7,200 points in his fantasy baseball league, which is 20% points than Adam. How many points does Adam have?
Adam has 6,000 points.
Explanation:Let's use algebra to solve this problem. Let's assume that Adam's points are represented by 'x'. According to the problem, Nick has 20% more points than Adam, so Nick's points can be represented as 'x + 20% of x' or '1.2x'. Given that Nick has 7,200 points, we can set up the equation 1.2x = 7,200 and solve for 'x'. Divide both sides of the equation by 1.2 to isolate 'x' and find that Adam has 6,000 points.
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Lawrence's parents pay him a base allowance of $20 per week and $3.55 per hour for extra chores he completes. Mrs. Johnson pays Lawrence $7.15 per hour to lifeguard at the city pool. Which equation models Lawrence's total weekly income? A. I = 7.15x ? 3.55y ? 20 c. I = (3.55 + 7.15)x + 20 b. I = (3.55 + 20)x + 7.15y d. I = 3.55x + 7.15y + 20
Answer:
7.15x + 3.55y + 20 = weekly income
Step-by-step explanation:
7.15x + 3.55y + 20 = weekly income
URGENT. Please help me!! 30 points :))
Answer:
<T = 83
Step-by-step explanation:
In a parallelogram <M = < P and < N = <T (opposite angles are congruent)
We also know that <M + <N = 180 ( consecutive angles are supplementary)
<M + <N = 180
6x+10 + 5x+10.5 = 180
Combine like terms
11x+20.5 = 180
Subtract 20.5 from each side
11x +20.5-20.5 = 180-20.5
11x = 159.5
Divide each side by 11
11x/11 = 159.5/11
x=14.5
We can find <N
<N = 5x+10.5
= 5(14.5)+105
=72.5 +10.5
= 83
<N = <T = 83
use the given information to prove that BC=DE
2. CD = CD , Reflexive Prop.
3. BC = DE, Subtraction Prop.
what is the perimeter of triangle with side lengths of 29, 15, and 4xy?
Answer:
P =44 + 4xy
Step-by-step explanation:
To find the perimeter of a triangle, add up the three sides
P = 29+15+4xy
P =44 + 4xy
A certain star is 1.135 × 10^14 km away from Earth. If light travels at 9.4607 × 10^12 km per year, how long will it take for light from the star to reach Earth?
Answer:
12 years will it take for light from the star to reach Earth.
Step-by-step explanation:
As per the given statement: A certain star is 1.135 × 10^14 km away from Earth. If light travels at 9.4607 × 10^12 km per year.
⇒Speed of light travel = [tex]9.4607 \times 10^{12}[/tex] km per year
and Distance of a certain star from the Earth = [tex]1.135 \times 10^{14}[/tex] km
To find how long will it take for light from the star to reach Earth.
Using Formula:
[tex]\text{Speed} = \frac{\text{Distance}}{\text{Time}}[/tex]
or
[tex]\text{Time} = \frac{\text{Distance}}{\text{Speed}}[/tex]
Substitute the given values we have;;
[tex]\text{Time} = \frac{1.135 \times 10^{14}}{9.4607 \times 10^{12}}[/tex]
Simplify:
Time = 11.9969981 year ≈ 12 years
therefore, 12 years will it take for light from the star to reach Earth.
Help w/ Geometry!!
Here is a geometrical statement: All squares are rectangles
Write the statement in if-then form:
What is the hypothesis of the statement?
What is the conclusion of the statement?
Answer:
If the shape is a square,then it is a rectangle.
Step-by-step explanation:
The hypothesis of the statement is the one which follows after if,
If the shape is a square.
The conclusion of the statement is the sentence that follows after then,
Then it is a rectangle.
The statement 'All squares are rectangles' translates into an if-then form as 'If a shape is a square, then it is a rectangle'. The hypothesis is 'a shape is a square' and the conclusion is 'it is a rectangle'.
Explanation:
The geometrical statement 'All squares are rectangles' can be translated into an if-then form as follows: 'If a shape is a square, then it is a rectangle'. In this statement, the hypothesis is 'a shape is a square' and the conclusion is 'it is a rectangle'. This means that if we start with a shape and it happens to be a square (which is our condition or hypothesis), then it must also necessarily be a rectangle (which is our end result or conclusion). This is because all the properties of a rectangle - having all angles being 90 degrees and opposite sides being equal - are also properties of a square.
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What is the domain of f/g, given f(x)=x+8 and g(x)=x-3?
Answer: Correct Option is "C"
( - ∞ , 3) U (3, ∞ )
Step-by-step explanation:
The function f/g is defined as
( x + 8) / (x -3)
Domain of the function is the set of values which the independent variable can assume.
Clearly in the above function x cannot assume the value 3, otherwise the function would become undefined.
So domain of the function is
( - ∞ , 3) U (3, ∞ )
Hope it helps.
Thank you.