If the two functions be f(x) = 3x² - 11x - 4 and g(x) = 3x + 1 then
(fg)(x) = f(x)g(x) = [tex]$$9x^3[/tex] - 30x² - 23x - 4
(f/g)(x) = f(x)/g(x) = x - 4
What is meant by function operation?
Function operations are the rules we use to solve functions. There is a specific way to deal with function addition, multiplication, and division.
Function composition is a mathematical operation that takes two functions f and g and produces a function h = g f such that h(x) = g(f(x)). The function g is applied to the result of applying the function f to x in this operation.
Let the two functions be f(x) = 3x² - 11x - 4 and g(x) = 3x + 1.
(fg)(x) = f(x)g(x)
= (3x² - 11x - 4) (3x + 1)
simplifying the above equation, we get
= 3x² [tex]*[/tex] 3x + 3x² [tex]*[/tex] 1 - 11x [tex]*[/tex] 3x - 11x [tex]*[/tex] 1 - 4 [tex]*[/tex] 3x - 4 [tex]*[/tex] 1
= [tex]$$9x^3[/tex] - 30x² - 23x - 4
Therefore, (fg)(x) = f(x)g(x) = [tex]$$9x^3[/tex] - 30x² - 23x - 4
(f/g)(x) = f(x)/g(x)
Where, f(x) = 3x² - 11x - 4 and g(x) = 3x + 1.
f(x)/g(x) = (3x² - 11x - 4)/(3x + 1)
= [(3x + 1)(x - 4)]/(3x + 1)
canceling the common factors we get
= x - 4
Therefore, f(x)/g(x) = x - 4
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| | x+4|-7 |< 5, | | = absolute value
The absolute value of the inequality is x < 6
What is absolute value?Absolute value can be defined as the describing the distance of a number from zero on the number line, without considering direction.
It is also the value of the number with no consideration to it's sign.
Given the inequality;
| | x+4|-7 |< 5 | |
Take the absolute value and expand the bracket
(x + 4) + 7 < 5
expand the bracket
x + 4 + 7 < 5
x + 11 < 5
Make 'x' the subject
x < 5 - 11
x < -6
Take the absolute value
x < 6
Thus, the absolute value of the inequality is x < 6
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Paris went to the store with $20. She bought a Gatorade for $2.50, and three packs of gum. If she leaves with
$12.43, how much was each pack of gum? Be sure to define a variable, and write an equation.
Answer:
Each pack of gum was 1.69
Step-by-step explanation:
20 - 2.50 = 17.50
17.50 - 12.43 = 5.07
5.07/3 = 1.69
Unit cube:
What is the volume of the following figure?
cubic units
Front View
Back View
Answer:
back view
Step-by-step explanation:
You are very very very correct
The total consumption of fruit juice in a particular country in 2006 was about 1.92 billion gallons. The population of that country in 2006 was 500 million. What was the average number of gallons of fruit juice consumed per person in the country in 2006? Using the per person amount from this problem, about how many gallons would your class consume? pls help
The average quantity of juice that is consumed per person in this country is given as 3.84
How to solve for the averageWe have the following data with which to answer this question
The consumption of juice = 1.92 billion
The population of the country = 500 million
The question wants us to calculate the amount of drink that the persons would consume on average in this nation.
we have average = quantity consumed / total number of persons
This would give us
average = 1.92 billion / 500 million
= 1, 920, 000, 000 / 500, 000 ,000
= 3.84
Hence the amount of juice that is consumed per person in this country is given as 3.84
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what is the vertex of this graph
(minimum or maximum)
Answer:
maximum at (0, 2 )
Step-by-step explanation:
the vertex is the turning point on the graph.
the graph increases on the left up to (0, 2 ) then decreases
this indicates the vertex is a maximum
f(a)= a +1
g(a)=-2a²-3
Find (f x g)(-4)
In the given expressions (f x g)(-4) is -8a³ - 8a² + 12a + 12.
What exactly are expressions?A mathematical expression is a sentence that has at least two variables or numbers and one or more mathematical operations.Mathematicians can multiply, divide, add, or subtract.An expression is structured as follows: Number/variable, mathematical operator, and expressionSo, (a+1)(2a²-3)(-4):
(a+1)(2a²-3)(-4)a(2a²-3) +1(2a²-3)(-4)(2a³-3a +2a²-3)-4-8a³ - 8a² + 12a + 12Therefore, in the given expressions (f x g)(-4) is -8a³ - 8a² + 12a + 12.
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What is each number in simplest form?
a. 32⁻³/⁵
Fractions are in their simplest form when they have no common divisors other than one at their top and bottom. The top and bottom cannot be divided further and remain integers.
You may also hear the simplest form called "lowest term."
In preferred, an expression is in only form while it's far easiest to use not unusual methods to help you simplify:
Combine Like termsFactoringincreasing (the opposite of factoring)clean out fractions by multiplyingrecognizing a pattern you have seen earlier than, just like the difference of squaresInput:
32^ (-3/5)
Exact Value:
1/8
Decimal Form:
0.125
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find four rational between -5/6 and -4/9
The four rational number between -5/6 and -4/9 are (-43/54, -37/54, -35/54 and -31/54).
First we will make the denominator of -5/6 and -4/9 same by multiplying the numerator and denominator of -5/6 by 9 and multiply the numerator and denominator of -4/9 by 6, we get
-5x9/6x9 = -45/54 and -4x6/9x6 = -24/54
Now, we have -45/54 and -24/54 whose denominator are same.
we can choose or pick any four integers as numerator in between -45 and -24 with denominator as 54.
-43/54, -37/54, -35/54 and -31/54 are the four rational number having numerator in between -45 and -24 with denominator as 54.
Therefore, -43/54, -37/54, -35/54 and -31/54 are the four rational number between -5/6 and -4/9.
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Sam will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of $65 and costs an additional $0.30 per mile driven. The
second plan has no initial fee but costs $0.80 per mile driven. How many miles would Sam need to drive for the two plans to cost the same?
Sam would need to drive for the two plans to cost the same is 130 miles.
Let, x = Number of miles driven
y = Total cost
The first plan:
Initial fee = $65
Cost of 1 mile = $0.3
Cost of drive x mile = $0.3x
In total, you will pay
y = 0.3x + 65 for the first plan.
The second plan:
There are no initial costs, but
Cost of 1 mile = $0.80
y = 0.80x
Now we have two equations
y = 0.3x + 65
y = 0.80x
As we know that y is equal to both 0.3x + 65 and 0.80x, this means we can equate the right hand sides and solve for x.
0.3x + 65 = 0.80x
0.80x - 0.3x = 65
0.5x = 65
x = 65÷0.5
x = 130
The cost of two plans will be same when Sam drive 130 miles.
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The sum of two consecutive numbers is 85, find the numbers
Answer:
85
Step-by-step explanation:
Consecutive numbers are numbers that follow each other in order. The two consecutive numbers are 42 and 43.
42+43=85
Answer:
42 & 43
Step-by-step explanation:
Let the two consecutive numbers be x and x+1 .
Here it is given that the sum of two consecutive numbers is 85.
So we get the equation,
x + (x+1) = 85
We get,
x + x + 1 = 85
2x + 1 = 85
Transposing +1 to RHS,
2x = 85 - 1
2x = 84
Transposing 2 to RHS,
x = 84 ÷ 2
Therefore, we get x = 42
Therefore, the two numbers are x and x+1, so they are: 42 and 43.
I hope this helped you! Pls mark me brainliest! Have a great day! <3
Write each equation in logarithmic form.
1/4=4⁻¹
The logarithmic form of 1/4 = 4⁻¹ is log₄ 4⁻¹ = ₋1.
The fundamental logarithmic function has the notation f(x) = log(r) y = log(x), where a > 0. The exponential function ay = x's inverse is represented by this. The common logarithm (log) or natural logarithm (ln) are examples of log functions (log).
The exponential function of the type aˣ = N can be converted into the logarithmic function logₐN = x. A Napier logarithm table can be used to determine the logarithmic value of any number. Logarithms are often calculated using a base of 10, therefore it is possible to get the logarithmic value of any number.
we have the exponential form as aˣ = N
1/4 = 4⁻¹
now the logarithmic form is:
logₐ N = x
log₄ 4⁻¹ = ₋1
hence we get the logarithmic form as log₄ 4⁻¹ = ₋1.
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help please 3 2/5 ⋅ 2 1/3
Answer:
7 14/15
Step-by-step explanation:
Round your answer to the nearest hundredth B C 7 A 65°
The value of BC the nearest hundredth is 15.01 .
Trigonometric Ratios area unit outlined because the values of all the pure mathematics functions supported the worth of the quantitative relation of sides in an exceedingly trilateral. The ratios of sides of a trilateral with relation to any of its acute angles area unit referred to as the pure mathematics ratios of that exact angle.The 3 sides of the correct triangle are:Hypotenuse (the longest side)Perpendicular (opposite aspect to the angle)Base (Adjacent aspect to the angle)From the image , we are able to see that CA = seven and angle A = 65°
Using ,Trigonometric Ratios , we get
Tan A = Perpendicular / Base
Tan 65° = BC / CA
2.144 =BC / 7
2.144 ×7 = BC
15.011 = BC
On rounding off to the closest hundredth , we have a tendency to get BC = 15.01
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f(x)=√x-4
y = √x=4
How to find the inverse function
f ⁻¹( x)= x^2+4 is the inverse of f(x)= [tex]\sqrt{x-4}[/tex]
what is an inverse function ?
An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by 'f' or 'F', then the inverse function is denoted by f-1 or F-1.
write f(x)= [tex]\sqrt{x-4}[/tex] as an equation
y= [tex]\sqrt{x-4}[/tex]
interchange the variables
x= [tex]\sqrt{y-4}[/tex]
solve the equation for y we get
squaring both side ,we have
x^2 = y-4
this implies that
y= x^2 +4
replace y with f ⁻¹( x) to show the final answer
f ⁻¹( x)= x^2+4
Hence, f ⁻¹( x)= x^2+4 is the inverse of f(x)= [tex]\sqrt{x-4}[/tex]
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solve this question
3x+4x=80,000
Answer:
11429 (estimated)
Step-by-step explanation:
1. Combine like terms: 3x+4x=7x
7x=80,000
2. Divide 80,000 by 7, which is 11428.57
which can be rounded to 11429
Simplify each expression. x³/₅ / x³/₁₀
The result of simplifying the expression x³/₅ / x³/₁₀ using the exponent rules is [tex]\sqrt[10]{}[/tex](x³)
To solve this exercise we have to resolve algebraic operations following the exponent rules.
[(x)³/₅ ] / [(x)³/₁₀]
Using the quotient rule that indicates that: the exponent result will be the subtraction of these exponents, we have:
x⁽³/⁵ ⁻³/¹⁰⁾
(x)⁽⁶⁻³⁾/¹⁰
(x)³/¹⁰
As we have a fractional exponent, you must convert the exponent to root:
[tex]\sqrt[10]{}[/tex](x³)
What is an exponent?In mathematics an exponent is the number of time that a number, called (base) is multiplied by itself. It is also called, power or index.
Example: 3² = 3*3 = 9
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For any function, we call x the ____ value and y is the ____ value.
For any function, we call x the numerator value and y is the denominator value.
What is a function?A function can be defined as a mathematical expression which is used to define and represent the relationship that exists between two or more variables. This ultimately implies that, a function is typically used for mapping an input variable to an output variable.
The parts of a fraction.In Mathematics, a fraction comprises two (2) main parts and these include the following:
NumeratorDenominatorFor any function in Mathematics, we call x the numerator value while y is the denominator value.
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Ranger wants to verify that the expression
8x - 20 is the correct simplified expression for 4(2x - 5) Explain how to verify the simplified expression.
Answer: It is correct.
The first step here is to distribute the numbers
4 times 2x is 8x
4 times -5 is -20
Therefore, you're left with 8x-20
I hope this helps!
Find the approximate length (in Earth years) of each planet's year.
b. A Jovian year if Jupiter is 5.46 AU from the sun.
The approximate length is 12.755 year or 4653.75 days.
A year is the time it takes for a planet, moon, or other celestial body to make one orbit around the Sun. ( when we take our solar system as an example). In the case of other exoplanets, the celestial bodies, moon and other planets orbit around the main star. The period of the moon's orbit around the earth is related to the moon.
The reference point is very important in determining the exact length of a year. The length is dependent upon the reference point used for the orbital movement.
Jupiter completes a Single orbit once every 4333 days, due to which one Jupiter year is equivalent to 11.86 earth years.
The complete calculation is done below. (Image Attached)
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1-(r+2)=9 (pls help, I'm not the smartest person)
Answer:
r = -10
Step-by-step explanation:
1. Distribute
1-(r+2)=9
1-r-2=9
2. Subtract the numbers
1-r-2=9
-1-r=9
3. Rearrange the terms
-1-r=9
-r-1=9
Solution: r = -10
Using the quadratic function represented by the graph below and answer the following
questions.
The vertex is (2,-3)So the Answer isY=(x-2)-3
Solve each equation.
2³x=4x⁺¹
Solution of exponential equation is , x=2
What is exponential equation ?In, mathematics a exponential equation or exponential function is inverse of logarithmic function. That means, we can easily convert the logarithmic function into exponential function and vice versa is also possible . The basic form of the exponential function can be written as .
[tex]a^n = x[/tex]
where, a = the base of function
x = main expression
n = exponential
Mathematically, if [tex]log_ax = n[/tex] is an logarithmic function then, this can be converted to exponential function as [tex]a^n = x[/tex] .
There are many formulas that are given to us for solving several simple and complex problems based on exponential functions and exponential equations.
According to given statement,
2^3x = 4^(x+1)
the equation can also written as :
[tex]= > 2^{3x} = 2^{2(x+1)}\\= > 3x = 2(x+1)\\= > 3x = 2x +2\\= > x=2[/tex]
Thus solution for this equation is x = 2.
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The complete question is :
Solve each equation.
[tex]2^{3x} = 4^{(x+1)}[/tex]
how do you write 84 with letters?
Answer:eighty four
Step-by-step explanation:
If a polynomial f(x) has a remainder of -6 when divided by x-5, what is f(5)?
Answer:
Step-by-step explanation:
The answer is just 5
f(5)=5
2y-10x=14 in slope intercept form
Answer:
You want to start off my isolating the y onto one side:
2y-10x=14 ---> 2y=14+10x
then you remove the coefficient from the left side:
2y=14+10x -----> y=7+5x
Then all you want to do is rearrange the equation to match slope intercept:
y=7+5x ----> y=5x+7
Answer:
You want to start off my isolating the y onto one side:
2y-10x=14 ---> 2y=14+10x
then you remove the coefficient from the left side:
2y=14+10x -----> y=7+5x
Then all you want to do is rearrange the equation to match slope intercept:
y=7+5x ----> y=5x+7
Step-by-step explanation:
What is the average rate of change of 2|x| on the interval [-3, 5].
The average rate of change of 2|x| on the interval [-3, 5] is 0.5
How to determine the average rate of change of 2|x| on the interval [-3, 5]?The function is given as
2|x|
Rewrite as
f(x) = 2|x|
The interval is given as
[-3, 5]
So, we have
f(-3) = 2|-3| = 6
f(5) = 2|5| = 10
The average rate of change of 2|x| on the interval [-3, 5] is then calculated as
Rate = (f(5) - f(-3))/(5 --3)
This gives
Rate = (10 - 6)/(5 --3)
Evaluate
Rate = 0.5
Hence, the average rate of change of 2|x| on the interval [-3, 5] is 0.5
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Will the answer to the addition problem below be odd or even?
742 +61 = ?
A. Odd
B. Even
Answer:
odd
Step-by-step explanation:
Not sure tho
What is the equation of the line that passes through the point (2,-1)and has a slope of 1
Answer:
y =x - 3
Step-by-step explanation:
Equation of a line: y = mx + bHere, m is the slope and b is the y-intercept.
slope = m = 1
y = 1x + b
Substitute the point (2,-1) in the above equation and find 'b'.
-1 = 1*2 + b
-1 = 2 + b
-1 -2 = b
b = -3
Equation of the line: y = x - 3
Fully simplify 7hw+5wh + 3hw
Answer:
15hw.
Step-by-step explanation:
Note 5wh is the same as 5hw, so we have:
7hw+5hw + 3hw
= 15hw.
Or, if you like, you could write it as 15wh.
By measuring the amount of carbon-14 in an object, a palaeontologist can determine its approximate age. The amount of carbon- 14 in an object is given by y=a e⁻⁰.⁰⁰⁰¹²t, where a is the amount of carbon- 14 originally in the object, and t is the age of the object in years. In 2003, a bone believed to be from a dire wolf was found at the La Brea Tar Pits. The bone contains 14 % of its original carbon-14. How old is the bone?
b. What properties of logarithms and exponents can you use to solve the equation?
The bone is approximately 16384 years old, as solved by basic algebra and logarithm.
The properties of logarithms and exponents so used are: [tex]log_b(a)^m=m(log_b(a))[/tex], [tex]log_e(e) = 1[/tex]
What is basic algebra and logarithm?Numbers, variables, constants, expressions, equations, linear equations, and quadratic equations are all part of the basics of algebra. Additionally, the algebraic expressions contain the basic arithmetic operations of addition, subtraction, multiplication, and division.This indicates that the exponent to which a fixed number, base b, must be raised in order to obtain a specific number x, is represented by the logarithm of that number.A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.Let 'a' be the original amount of the object.
Then, according to question, for y = amount of carbon in the object, y = 0.14(a) (as the bone contains 14% of its original carbon-14).
We are also given that: y = a [tex]e^{-0.00012t}[/tex]
On substituting the value of y, we get: 0.14a = a [tex]e^{-0.00012t}[/tex]
By basic algebra, we can solve it as:
=> 0.14 = [tex]e^{-0.00012t}[/tex]
=> ln (0.14) = ln ([tex]e^{-0.00012t}[/tex]) (taking natural logarithm on both sides)
=> ln (0.14) = - 0.00012t (as [tex]log_b(a)^m=m(log_b(a))[/tex]; further [tex]log_e(e) = 1[/tex])
=> [tex]\frac{ln(0.14)}{-0.00012}[/tex] = t
=> t = 16384 years (approx.)
Hence, the bone is approximately 16384 years old, as solved by basic algebra and logarithm; The properties of logarithms and exponents so used are: [tex]log_b(a)^m=m(log_b(a))[/tex], [tex]log_e(e) = 1[/tex].
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