Answers
Answer: No, they are not inverses of one another.
=======================================================
Explanation:
There are at least 3 methods we can use to prove the functions aren't inverses of each other.
------------------------------------
Method 1
Let's say we plugged x = 0 into f(x).
[tex]f(\text{x}) = 2\text{x}+1\\\\f(0) = 2*0+1\\\\f(0) = 1\\\\[/tex]
The input x = 0 leads to the output y = 1.
Then let's treat y = 1 as the input of the g(x) function, so we'll plug x = 1 into here.
[tex]g(\text{x}) = -2\text{x}-1\\\\g(1) = -2*1-1\\\\g(1) = -3\\\\[/tex]
If g(x) was the inverse of f(x), then the g(1) output value should be 0 to get us back where we started. However, we get an output of -3 instead.
This is sufficient evidence to conclude that f(x) and g(x) are not inverses of each other. We only need one counter-example to disprove this claim.
------------------------------------
Method 2
Recall that if f(x) and g(x) are inverses of each other, then we have the following two properties
f(g(x)) = xg(f(x)) = xThe left hand side of each equation involves the concept of "function composition".
Let's see what f(g(x)) is equal to in this case.
[tex]f(\text{x}) = 2\text{x}+1\\\\f(g(\text{x})) = 2(g(\text{x}))+1\\\\f(g(\text{x})) = 2(-2\text{x}-1)+1\\\\f(g(\text{x})) = -4\text{x}-2+1\\\\f(g(\text{x})) = -4\text{x}-1\\\\[/tex]
We do not get an output of simply x only, so this is another way to see how we don't have inverses.
Through similar steps, you should find that [tex]g(f(\text{x})) = -4\text{x}-3[/tex] which is further confirmation that f(x) and g(x) are not inverses of each other.
------------------------------------
Method 3
Let's ignore g(x) for now. We'll start with f(x) and find the inverse of it.
The process involves this basic outline:
Replace f(x) with ySwap x and ySolve for ySo,
[tex]f(\text{x}) = 2\text{x}+1\\\\\text{y} = 2\text{x}+1\\\\\text{x} = 2\text{y}+1\\\\\text{x}-1 = 2\text{y}\\\\\text{y} = \frac{\text{x}-1}{2}\\\\f^{-1}(x) = \frac{\text{x}-1}{2}\\\\[/tex]
Unfortunately we do not end up with -2x-1 as the actual inverse.
So if [tex]g(\text{x}) = \frac{\text{x}-1}{2}[/tex] was the case, then f(x) and g(x) would be inverses of each other.
Through similar steps, the inverse of [tex]g(\text{x}) = -2\text{x}-1\\\\[/tex] is [tex]g^{-1}(\text{x}) = -\frac{\text{x}+1}{2}\\\\[/tex]
Related Questions
you are to build a rectangular pen with three parallel partitions using 500 feet of fencing. what dimensions will maximize the total area of the pen?
Answers
The dimensions required are 125 feet and 50 feet.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.
The total length will be x and the height will be y.
Perimeter = 500 = 2x + 5y
Total Area = A = xy
Solve for y using the equation of perimeter:
5y = 500 - 2x
y = 100 - 2x/5
Put y in the equation of the Area
A = x(100 - 2x/5)
A = -2x²/5 +100x
To find the maximum area, find the derivative:
A' = -4x/5 + 100
To maximise the area, we equate A' to zero.
A' = 0
-4x/5 +100 = 0
∴ x = 125
A' is positive when x < 125 and A' is negative when x >125, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.
Put x = 125 in equation of perimeter:
500 = 2(125) + 5y
500 = 250 + 5y
5y = 250
∴ y = 50 ft
Thus, the dimensions required are 125 feet and 50 feet.
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Pls Help Got to do this right now!!!
Which expression is the additive inverse of - 13/6?
A) 13/6
B) 6/13
C) - 6/13
D) - 13/6
WILL MARK BRAINLIEST!!
Answers
Explanation- inverse mean opposite
Rewrite 15x^4y^3 − 45x^3y^4 using a common factor
Answers
15x^4y^3 − 45x^3y^4 Can be rewritten using a common factor as 15x^3y^3(x - 3y)
What is a common factor?common factors are defined as factors that are common to two or more numbers. In other words, a common factor is a number with which a set of two or more numbers will be divided exactly.
In the algebraic expression 15x^4y^3 − 45x^3y^4,
15x^4y^3 is a term and 45x^3y^4, is another term.
if we take each variable and constant and compare
15 is common factor between 15 and 45
x^3 is a common factor between x^4 and x^3
y^3 is a common factor between y^3 and y^4
so multiplying the common factors together we have 15x^3y^3
what is left in the first term is x and second term is 3y
so that the rewritten form becomes 15x^3y^3(x - 3y)
In conclusion the other form of 15x^4y^3 − 45x^3y^4 is 15x^3y^3(x - 3y)
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How much compound interest will $50,000 have earned in 10 years at
6.4% annual interest compounded quarterly?
Answers
The compound interest earned on $50,000 in 10 years at 6.4% annual interest compounded quarterly would be $28,167.50.
What is Compound interest?Compound interest is a method of calculating the interest charge. In other words, it is the addition of interest on interest.
If the initial amount (also called as principal amount) is P, and the interest rate is R% per unit time, and it is left for T unit of time for that compound interest, then the interest amount earned is given by:
Compound Interest =P(1+r/n)^rt
We are given that
Principal amount = $50,000
Time = 10 year
Rate = 6.4%
Determine the compound interest earned on $50,000 in 10 years at 6.4% annual interest compounded quarterly, we can use the formula:
Plugging in the given values, we get:
compound interest = $50,000 (1 + 0.064/4)^(4 x 10) - $50,000
= $50,000 (1.016)^40 - $50,000
= $50,000 x 2.6335 - $50,000
= $78,167.50 - $50,000
= $28,167.50
Therefore, the compound interest is $28,167.50.
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(10)
Write the equation
of the line in
Slope-Intercept
Form.
Answers
Answer:
y = -x
Step-by-step explanation:
The slope of this line is -1 (rise/run = -1/1 = -1). We can plug this value into the slope-intercept form equation:
y = mx + b
y = -1x + 0
and simplify.
y = -x
Ru buy three cake for £1. 60 each. He pay with a £20 note. How much change doe he get
Answers
We know that Ru will get back £15.2 when she buys 3 cakes for £1.60 each using simple mathematical operations.
What are mathematical operations?The process of calculating a value using operands and a math operator is known as a mathematical "operation."
The provided operands or integers must follow certain established rules that are associated with the math operator's symbol.
The rules that determine how to solve an expression with several operations are referred to as the order of operations.
PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction (from left to right).
So, we know that:
Ru buys 3 cakes for £1.60 each.
She was given £20 not.
The money she will get back is:
20 - (1.60*3)
20 - 4.8
£15.2
Therefore, we know that Ru will get back £15.2 when she buys 3 cakes for £1.60 each using simple mathematical operations.
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What is the distance from the origin to pint A graphed on the complex plane below
Answers
Answer: it's [tex]\sqrt{5}[/tex]
Step-by-step explanation:
8. Graph the function
f(x)=-5/4x+6.
Label the point on the graph that is the
solution to f(x) = -4 by writing
solution" next to it. Then, write the
solution below.
Solution
Show your work
Answers
The graph for the given function f(x)=-5/4x+6. is obtained.
What is meant by the term function?A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.For the stated question, the function is defined as;
f(x) = -5/4x + 6.
Put x = 0,
f(0) = -5/4(0) + 6.
f(0) = 6
Coordinate (0, 6)
For, the solution of f(x) = -4, put f(x) = -4
-4 = -5/4x + 6
x = 8
Coordinate (8, -4)
Plot the graph using the obtained points for the function.
Thus, the graph for the given function is obtained.
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What is the value of g(5)-2g(1)
Answers
Answer:
3 grams
Step-by-step explanation:
g(5)-2g(1)
g × 5- 2g(1)
5g - 2g×1
5g - 2g
3g
Two piece of metal meaure one and one fifth yard and three tenth yard each. Three time the amount of metal i needed for a project. How many total yard of metal i needed? one and one half yard one and nine fifteenth yard four and one half yard four and one fifth yard
Answers
As per the area of the rectangle, the amount of metal need for the total yard is 4.68
Area of rectangle
In geometry, the area of rectangle is written as,
A = l x b
where l refers the length and b refers the breadth of the rectangle.
Given,
Here we have given that the 1 1/5 and 3 1/10 is the measure of yard.
While we looking into the given question, we have identified that,
3 time of metal is need for the project,
So, as per the area of the rectangle formula, it can be written as,
=> 3 x 1 1/5 x 3 1/10
=> 3 x 6/5 x 13/10
=> 234/50
=> 4.86
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Answer:
4 1/2
Step-by-step explanation:
add 1 1/5 and 3/10 and then multiply your answr by 3
Given: ∠BCD is right; BC ≅ DC; DF ≅ BF; FA ≅ FE
Triangles A C D and E C B overlap and intersect at point F. Point B of triangle E C B is on side A C of triangle A C D. Point D of triangle A C D is on side C E of triangle E C D. Line segments B C and C D are congruent. Line segments B F and F D are congruent. Line segments A F and F E are congruent.
Which relationships in the diagram are true? Select three options.
△ACF ≅ △ECF by HL
ΔCBF ≅ ΔCDF by SSS
ΔBFA ≅ ΔDFE by SAS
ΔCFD ≅ ΔEFD by SSS
ΔCBE ≅ ΔCDA by HL
SAS
Answers
The three relationships in the diagram that are true include the following;
B. ΔCBF ≅ ΔCDF by SSS
C. ΔBFA ≅ ΔDFE by SAS
E. ΔCBE ≅ ΔCDA by HL
What is the Right Triangles Similarity Theorem?In Mathematics, the Right Triangles Similarity Theorem states that when the altitude of a triangle is drawn to the hypotenuse of a right angled triangle, then, the two (2) triangles that are formed would be similar to each other, as well as the original triangle.
Based on the Right Triangles Similarity Theorem, we have the following;
ΔCBF ≅ ΔCDF by side, side, side (SSS).ΔBFA ≅ ΔDFE by side, angle, side (SAS).What is HL?In Mathematics, HL is an abbreviation for hypotenuse leg and it can be defined as a theorem which states that if the hypotenuse and one (1) leg in a right-angled triangle are congruent to the hypotenuse and leg of another right-angled triangle, then the two (2) triangles would be congruent.
By applying the hypotenuse leg (HL) theorem across right-angle BCD (∠BCD), we have the following;
ΔCBE ≅ ΔCDA
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Average employee salary?
Answers
Answer:
$67,680
Step-by-step explanation:
average is calculated as
average = [tex]\frac{sum}{count}[/tex]
Denver
[tex]\frac{sum}{137}[/tex] = 67,013 ( multiply both sides by 137 )
sum = 9,180,781
Anchorage
[tex]\frac{sum}{25}[/tex] = 71,332 ( multiply both sides by 25 )
sum = 1,783,300
total sum for both = $9,180,781 + $1,783,300 = $10,964,081
then average salary across both companies is
average = [tex]\frac{10,964,081}{162}[/tex] = $67,680 ( to the nearest dollar )
Cindy had $660 in a savings account when Erika opened a savings account with zero dollars.
Cindy deposited $60 into her account each month for x months.
Erika deposited $100 into her account each month for x months.
• The accounts did not earn interest.
Which inequality represents this situation when the amount of money in Cindy's account was greater than the amount of money in Erika's accour
Answers
Answer: To represent the situation where the amount of money in Cindy's account is greater than the amount of money in Erika's account, you can set up the inequality as follows:
Cindy's account balance + (Cindy's monthly deposit * number of months) > Erika's account balance + (Erika's monthly deposit * number of months)
Substituting the known values, we get:
660 + (60 * x) > 0 + (100 * x)
Combining like terms, we get:
660 + 60x > 100x
Subtracting 60x from both sides, we get:
660 > 40x
Dividing both sides by 40, we get:
16.5 > x
Therefore, the inequality that represents the situation when the amount of money in Cindy's account is greater than the amount of money in Erika's account is 16.5 > x. This means that for the number of months (x) to be greater than 16.5, the amount of money in Cindy's account must be greater than the amount of money in Erika's account.
7 divided by 1 over 2
Answers
Answer: 3.5
Step-by-step explanation: hope it helps :)
Answer: 14
Step-by-step explanation: using cross multiplication, 7 times 2=14 and 1 time 1=1. so it would be 14/1 which can be simplified to 14.
Lester bought 8 t-shirts. The shirts came in p packages. Write an expression that shows how many t-shirts were in each package.
Answers
Expression for Number of t shirts in each package = [tex]\frac{8}{p}[/tex]
What is Unitary Method ?The unitary approach involves calculating the value of a single unit, from which we can calculate the values of the necessary number of units.
Total number of shirts bought = 8 T- shirts
Total number of packages = p
So,
According to unitary method
In p packages we have t shirts = 8
In one package number of t shirts we will have = [tex]\frac{8}{p}[/tex]
Expression for Number of t shirts in each package = [tex]\frac{8}{p}[/tex]
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Dylan has $14.00 to spend at the book fair. He decides to buy two paperbacks for $4.50 each. Then, he sees some sticker books. At most, how much can he spend on sticker books?
A $5.50
B $6.00
C $5.01
D $5.00
Answers
Answer:
B because I had learned this at school.
The diameter of a circle is 15 cm. Find its area to the nearest whole number.
Answers
Answer:
177
Step-by-step explanation:
A=3.14r^2
A=3.14(7.5)
A=~177
given a fixed point and a fixed line, a parabola consists of all points that are equidistant to the fixed point and the fixed line. State True or False your answer:
a. True
b. False
Answers
It is true that given a fixed point and a fixed line, a parabola consists of all points that are equidistant to the fixed point and the fixed line.
A parabola is a quadratic function graph .A parabola is a curve equation in which a point on the curve is equidistant from a fixed point and a fixed line.
The fixed point is known as the parabola's focus, and the fixed line is known as the parabola's directrix. It is also worth noting that the fixed point does not lie on the fixed line.
A parabola is a locus of any point that is equidistant from a given point (focus) and a given line (directrix). The parabola is a significant curve of the coordinate geometry's conic sections.
The general equation of a parabola is: y = a(x-h)² + k or x = a(y-k)² +h, where (h,k) denotes the vertex.
The standard equation of a regular parabola is y² = 4ax.
It's true that given a fixed point and a fixed line, a parabola consists of all points that are equidistant to the fixed point and the fixed line.
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Find the measure of angle f given the information below. please help !!
Answers
The measure of Angle f from the the figure is [tex]53^{o}[/tex].
What is the relation between Angles and Parallel lines ?On a shared plane, parallel lines do not cross one another. Now, if a transversal crosses two parallel lines at two separate sites, four angles are created at each location. The characteristics of parallel lines with respect to transversals are therefore listed below.
Angles that coincide are equal.Angles that are vertically opposite or vertically angled are equal.Interior angles that alternate are equal.Exterior angles that alternate are equal.On the same side of the transversal, a pair of internal angles are additional.The pair of alternative internal angles are equal when two parallel lines are crossed by a transversal.
When a transversal cuts between two parallel lines, the interior angles on the same side of the transversal are additional.
In the problem angle
a = [tex]119^{o}[/tex]
e = [tex]66^{o}[/tex]
Now , c = e ( alternate angles, interior)
⇒ c = [tex]66^{o}[/tex]
a + b = [tex]180^{o}[/tex] ( as sum of angles on a straight line is [tex]180^{o}[/tex] )
b = [tex]180^{o}[/tex] - [tex]119^{o}[/tex] = [tex]61^{o}[/tex]
f + b+ c = [tex]180^{o}[/tex] [sum of angles of a Triangle is [tex]180^{o}[/tex] ]
f = [tex]180^{o}[/tex] - [tex]61^{o}[/tex] - [tex]66^{o}[/tex] = [tex]53^{o}[/tex].
The measure of Angle f from the the figure is [tex]53^{o}[/tex].
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PLEASE I NEED HELP!!!!!!!!
Answers
Answer:
161 square inches.
Step-by-step explanation:
The surface area of a pyramid is the sum of the areas of all of its faces. Since the base of the pyramid is a square with an edge length of 7 inches, the area of the base is 49 square inches (7 inches x 7 inches).
To find the surface area of the pyramid, you also need to find the area of the lateral faces of the pyramid. The lateral faces of a pyramid are the triangular faces that are not the base. The lateral faces of a pyramid are all congruent, meaning that they are all the same size and shape.
To find the area of a lateral face of the pyramid, you can use the formula for the area of a triangle: A = 1/2bh, where A is the area of the triangle, b is the base of the triangle, and h is the height of the triangle.
In this case, the base of the triangle is 7 inches (the edge length of the base of the pyramid) and the height of the triangle is 8 inches (the slant height of the pyramid). The area of each lateral face of the pyramid is therefore 1/2 x 7 inches x 8 inches = 28 square inches.
Since there are 4 lateral faces on the pyramid, the total area of the lateral faces is 4 x 28 square inches = 112 square inches.
To find the surface area of the pyramid, add the area of the base to the area of the lateral faces: 49 square inches + 112 square inches = 161 square inches. Therefore, the surface area of the pyramid is 161 square inches.
A new Youth Activity Center is being built in Ook Valley. The perimeter of the rectangular playing field is 272 yards. The length of the field is 8 yards less than double the width. What
are the dimensions of the playing field?
Answers
Answer:
w = 48 and l = 88
Step-by-step explanation:
A new Youth Activity Center is being built in Ook Valley. The perimeter of the rectangular playing field is 272 yards. The length of the field is 8 yards less than double the width. What are the dimensions of the playing field?
P = 2w + 2l
w = w
l = 2w - 8
272 = 2w + 2(2w - 8)
272 = 2w + 4w - 16
272 = 6w - 16
288 = 6w
w = 48
remember that: l = 2w - 8
l = 2w - 8 when w = 48
l = 2(48) - 8
l = 96 - 8
l = 88
CHECK: w = 48 and l = 88
P = 2w + 2l
272 = 2(48) + 2(88)
272 = 96 + 176
272 = 272
so: w = 48 and l = 88
when the philies play the dodgers, the tickets for three adults and three children are $75. for two adults and four children the total admission cost is $62. find the cost of each adult and each child ticket.
Answers
when the philies play the dodgers, the tickets for three adults and three children are $75. for two adults and four children the total admission cost is $62. So, by using equation formula:
the cost of each adult is $19
and each child ticket is $6
What is an equation?An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used.
The relationship between the two expressions on each side of the sign is represented by a mathematical equation. One variable and an equal sign are often present.
Let us assume,
tickets for adults = a
tickets for children = c
Now,
3a + 3c = $75 ........ (i)
2a + 4c = $62 ......... (ii)
Multiply the equation (i) ×2 and equation (ii) × 3, we get -
6a + 6c = $150 ........ (iii)
6a + 12c = $186 .........(iv)
Now, subtract equation (iv) from (iii) we get -
or, 6c = $36
or, c = $6
Now, putting the value of c in equation (iii) we get -
6a + 6×6 = $150
6a = $114
a = $19
Thus, the cost of each adult is $19
and each child ticket is $6
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A cashier gets a $100 bonus for working on a holiday plus $11/h. The total holiday salary is given by the function y = 11x + 100. How will the graph change if the bonus is raised to $150 and the hourly rate is lowered to $10/h? Choose the answers that describe these transformations.
HELP PLEASE
Answers
(50,650) the graph evolve if the hourly rate is reduced to $10/h and the incentive is increased to $150.
Given that,
For working on a holiday, a cashier is given a $100 bonus in addition to $11/h. The equation y = 11x + 100 yields the entire holiday salary.
We have to find how would the graph evolve if the hourly rate is reduced to $10/h and the incentive is increased to $150.
We know that,
We get,
A cashier is given a $100 bonus in addition to $11/h is y=11x+100
The hourly rate is reduced to $10/h and the incentive is increased to $150 is y=10x+150
Now we subtract those two equation
11x+100-10x-150=0
x-50=0
x=50
We get y as y=10x+150
y=10(50)+150
y=500+150
y=650
Therefore, (50,650) the graph evolve if the hourly rate is reduced to $10/h and the incentive is increased to $150.
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There are 4 chocolate chip cookies and 3 oatmeal cookies in a cookie jar. If a child randomly takes 2 cookies, what is the probability that both are chocolate chip?
Answers
hope you have a great day!
50 POINTS IF ANSWERED PLS!!!!
Which angle is the biggest based on the sides of the triangles below? Explain your reasoning.
Answers
Answer:
angle c
Step-by-step explanation:
since the largest angle will be across from the longest side, angle c is the right answer because it is across from the longest side which measures 35 cm.
hope this helps
help due tomm!!! suppppper easyy!!
Answers
Answer: b/-3<a/-3
Step-by-step explanation:
a/-3<b/-3
b/-3<a/-3
What is the equation of the blue line
Answers
Answer:
y = -1/2 x + 2
Step-by-step explanation:
A (0,2)
B (4,0)
(y-2)/(0-2) = (x-0)/(4-0)
(y-2)/-2 = x/4
(-y+2)/2 = x/4
4 * (-y+2)/2 = x/4 * 4
-2y+4 = x
-2y = x -4
2y = -x + 4
y = - 1/2 x + 4/2
y = -1/2 x + 2
If 8% of a liquid is 56 mL, what is the whole
amount?
Please help
Answers
Answer:
700 mL
Step-by-step explanation:
You want to know the whole amount if 8% of it is 56 mL.
RelationThe problem statement tells you ...
8% × whole = 56 mL
Divide by the coefficient ...
whole = (56 mL)/(0.08) = 700 mL
The whole amount is 700 mL.
Your grocery bill before taxe wa $265. 24. You have to pay 7. 3 ale tax. How much i your grocery bill?
Answers
If the grocery bill before tax is $265.24 and you have to pay 7.3% tax, then he has to pay $284.6 for the grocery bill
The amount of the grocery bill before tax = $265.24
The percentage of tax = 7.3%
The amount of tax = The price of the grocery bill before tax × (7.3/100)
Substitute the values in the equation
= 265.24 × (7.3/100)
= 265.24 × 0.073
= $19.36
The final bill including tax = The price of the grocery bill before tax + The amount of tax
Substitute the values in the equation
= 265.24 + 19.36
= $284.6
Therefore, the grocery bill including tax = $284.6
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write an eqation of the line that passes throught the points (0,-2), (4,-2),
Answers
The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
The equation of the line is y = -2.
What is an equation of a line?The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
Example:
The slope of the line y = 2x + 3 is 2.
The slope of a line that passes through (1, 2) and (2, 3) is 1.
We have,
The equation of the line is given as,
y = mx + c
The line passes through the points (0, -2) and (4, -2).
Now,
Slope = m
m = (-2 + 2) / (4 - 0)
m = 0 / 4
m = 0
Now,
(x, y) = (0, -2)
y = mx + c
-2 = 0 x 0 + c
c = -2
Now,
m = 0 and c = -2
y = 0x -2
y = -2
Thus,
The equation of the line is y = -2.
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