Consider the functions f(x) = (5x +4)/(X+9) and g(x) = (9x – 4)/(5-x) (a) Find f(g(x)) (b) Find g(f(x)). (c) Determine whether the functions f and g are inverses of each other
Answers
To find f(g(x)), we first need to substitute g(x) into f(x):
f(g(x)) = (5g(x) + 4)/(g(x) + 9)
Now we substitute the expression for g(x):
f(g(x)) = (5(9x-4)/(5-x) + 4)/((9x-4)/(5-x) + 9)
We can simplify this expression by finding a common denominator for the fractions in the numerator and denominator:
f(g(x)) = ((45x - 20 + 20 - x)/(5-x))/((9x - 4 + 45 - x)/(5-x))
f(g(x)) = (44x + 25)/(54 - 8x)
To find g(f(x)), we substitute f(x) into g(x):
g(f(x)) = (9f(x) - 4)/(5 - f(x))
Now we substitute the expression for f(x):
g(f(x)) = (9(5x + 4)/(x + 9) - 4)/(5 - (5x + 4)/(x + 9))
We can simplify this expression by finding a common denominator for the fractions in the numerator and denominator:
g(f(x)) = ((45x + 36 - 4(x + 9))/(x + 9))/((45 - 5x + 4(x + 9))/(x + 9))
g(f(x)) = (41x + 288)/(41 - x)
To determine whether f and g are inverses of each other, we need to show that f(g(x)) = g(f(x)) = x. Let's first check if f(g(x)) = x:
f(g(x)) = (44x + 25)/(54 - 8x) ≠ x
Therefore, f and g are not inverses of each other.
We can also check if g(f(x)) = x:
g(f(x)) = (41x + 288)/(41 - x)
Multiplying the numerator and denominator by -1, we get:
g(f(x)) = (-41x - 288)/(x - 41)
g(f(x)) = -1(x + 288)/(x - 41)
g(f(x)) = -1(288 - 41x)/(41 - x)
g(f(x)) = x
Therefore, f and g are not inverses of each other.
(a) To find f(g(x)), we need to substitute g(x) into f(x):
f(g(x)) = f((9x - 4) / (5 - x))
= (5((9x - 4) / (5 - x)) + 4) / (((9x - 4) / (5 - x)) + 9)
= (5(9x - 4) + 4(5 - x)) / (9x - 4 + 9(5 - x))
(b) To find g(f(x)), we need to substitute f(x) into g(x):
g(f(x)) = g((5x + 4) / (x + 9))
= (9((5x + 4) / (x + 9)) - 4) / (5 - ((5x + 4) / (x + 9)))
= (9(5x + 4) - 4(x + 9)) / (5(x + 9) - (5x + 4))
(c) To determine if f(x) and g(x) are inverses, we need to check if f(g(x)) = x and g(f(x)) = x. Since f(g(x)) and g(f(x)) in parts (a) and (b) are not simplified to x, f(x) and g(x) are not inverses of each other.
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Related Questions
The supplement of an angle is 4 times as large as the angle. Find the angle
Answers
Answer:
mark me barinlest
Step-by-step explanation:
supplement angles have sum = 180, complement angles have sum = 90
180-x = 4(90-x)
180-x = 360 -4x
3x = 180
x = 180/3
x = 60
answer measure of the angle = 60
Answer:
36°Step-by-step explanation:
Supplementary angles sum to 180°.
Let the angle be x then the other angle is 4x.
Their sum is:
x + 4x = 1805x = 180x = 180/5x = 36A number pyramid with faces 1-4 is tossed and a spinner a-f is spun. find p( even and vowel)
a: 1/12 b: 1/6 c: 1/4 d: 1/3
Answers
The probability of getting even and vowel is 1/6 if the number pyramid with faces 1-4 is tossed and a spinner a-f is spun option (b) is correct.
What is probability?It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words the probability is the number that shows the happening of the event.
As we know:
When events are independent with and statement:
P(A and B) = P(A)×P(B)
Let's suppose P(A) = probability of getting even number
Which is P(A) = 2/4 = 1/2
P(B) = Probability of getting vowel
Which is P(B) = 2/6 = 1/3
P(A and B) = (1/2)(1/3) = 1/6
Thus, the probability of getting even and vowel is 1/6 if the number pyramid with faces 1-4 is tossed and a spinner a-f is spun.
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If x is an integer, which is the solution set of
-1 ≤ x < 2?
A.
C.
{0,1}
{0, 1,2}
B.
D.
{-1,0, 1, 2}
{-1,0, 1}
Answers
Answer:
D.
Step-by-step explanation:
{-1,0,1}
pls mrk me brainliest i need to get the next rank
Select the correct answer. Consider the graph of the function f(x) = e^x.
What is the y-intercept of function g if g(x) = 2f(x)+1 ?
A. (0,2)
B. (0,-1)
C. (0,3)
D. (0,1)
Answers
Taking into account the definition of y-intercept, the y-intercept of function g is (0,3) (option C).
What is y-interceptTo find the y-intercept of a function, simply replace x with 0; and then solve the equation.
That is, to find the intersection point with the ordinate axis, f(0) is calculated by making x = 0.
The y-intercept of g(x) = 2f(x)+1In this case, you know that:
f(x) = eˣ
g(x) = 2f(x)+1
Then, substituting the expression of the function f(x) in the function g(x), you obtain g(x)=2×eˣ + 1
The y-intercept of function g is g(0). So, g(0)=2×e⁰ + 1
Solving:
g(0)=2×1 + 1
g(0)= 2+1
g(0)= 3
Finally, the y-intercept of function g is (0,3) (option C).
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When the first letter of a variable name is uppercase, as in hourlywage, the format is known as?
Answers
5x-4y=-18 and x-4y=6
Answers
Answer:
x = -6 and y = -3
Step-by-step explanation:
x- 4y =6
x =4y +6......(i)
5x -4y =-18
20y +30 - 4y = -18
16y = -48
y = -3
x = -12+6
x = -6
Is there another way to convert fractions to decimals (explanation pls ❤️)
Answers
Answer:
The easiest way to convert a fraction to a decimal is to divide the numerator (the top of the fraction) by the denominator (the bottom of the fraction) by using a calculator. The resulting answer will be the value of the fraction expressed as a decimal number.
Step-by-step explanation:
Answer: The simplest method is to use a calculator.
Just divide the top of the fraction by the bottom, and read off the answer. Hope this helps! If not, I’ll come up with another explanation! :)
N
Select the correct answer from the drop-down menu.
Which equation satisfies all three pairs of a and b values listed in the table?
a b
0-10
1
-7
2 -4
The equation is?
Answers
Answer:
An equation that satisfies all three pairs of a and b values listed in the table include the following: C. 3a - b = 10
Step-by-step explanation:
How to determine an equation that satisfies all three pairs of a and b values listed in the table?
In order to determine an equation that satisfies all three pairs of a and b values listed in the table, we would substitute each of the numerical values corresponding to each variable into the given equations and then evaluate as follows;
a - 3b = 10
0 - 3(-10) = 30 (False).
3a + b = 10
3(0) - 10 = -10 (False).
3a - b = 10
3(0) - (-10)
0 + 10 = 10 (True).
3a - b = 10
3(1) - (-7)
3 + 7 = 10 (True).
3a - b = 10
3(2) - (-4)
6 + 4 = 10 (True)
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Complete Question:
Which equation satisfies all three pairs of a and b values listed in the table?
a b
0 -10
1 -7
2 -4
The equation is?
A.) a-3b=10
B.) 3a+b=10
C.) 3a-b=10
D.) a+3b=10
| a | b |
| --- | --- |
| 0 | -10 |
| 1 | -7 |
| 2 | -4 |
We want to find an equation that satisfies all three pairs of a and b values. Let's first solve for b by substituting the given values for a and b into the equation:
b = -a^2 + a - k
0 = -10^2 + 10 - k
0 = 100 + 10 - k
-110 = -k
k = 110
Plugging k into the equation, we get:
b = -a^2 + a - 110
Is this the equation we're looking for? To find out, let's substitute the given values for a and b in the equation and see if it matches:
b = -0^2 + 0 - 110
b = -0 + 0 - 110
b = -110
b = -7
Yes, this equation satisfies all three pairs of the given a and b values! So our final answer is:
b = -a^2 + a - 110
We can use this equation to find the value of b given any value of a between 0 and 10.
Let A={(x-3)/(x-2)ЄR : X<0}
be a subset of real numbers.
i) Define A's supremum and infimum.
Answers
The supremum of the set A does not exist (it is negative infinity), and the infimum of the set A is 1.
To define the supremum and infimum of the set A, we first need to determine the properties of the set.
The set A is defined as A = {(x-3)/(x-2) ∈ R : x < 0}.
To find the supremum (also known as the least upper bound) of A, we need to find the smallest value that is greater than or equal to all the elements of A. In other words, we are looking for the least upper bound of the set A.
Let's analyze the elements of A:
For x < 0, the expression (x-3)/(x-2) can take on different values depending on the value of x. We need to find the maximum value that this expression can reach for all x < 0.
As x approaches 0 from the left side, (x-3)/(x-2) approaches negative infinity. Therefore, there is no finite supremum for the set A.
Next, let's find the infimum (also known as the greatest lower bound) of A. We need to find the largest value that is less than or equal to all the elements of A. In other words, we are looking for the greatest lower bound of the set A.
Again, analyzing the elements of A:
As x approaches negative infinity, (x-3)/(x-2) approaches 1. Therefore, the infimum of the set A is 1.
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Square has side length 2. the diagonal is approximately how long?
Answers
julian rolled a normal 6-sided die 12 times. his rolls were as follows: 2, 4, 3, 3, 5, 1, 2, 6, 3, 1, 3, 5, 4. what is the probability that he will roll a 3 on the next roll?
Answers
The probability that Julian will roll a 3 on the next roll is approximately 16.67%. The probability of rolling a 3 on a normal 6-sided die is independent of the previous rolls. This means that regardless of the outcomes of Julian's previous rolls, the probability remains the same.
Explanation
On a 6-sided die, there is 1 favorable outcome for rolling a 3 (the number 3 itself) out of 6 possible outcomes (1, 2, 3, 4, 5, and 6).
To find the probability, you can use the formula:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
In this case:
Probability of rolling a 3 = 1 (favorable outcome) / 6 (total outcomes)
Probability of rolling a 3 = 1/6 ≈ 0.1667 or 16.67%
So, the probability that Julian will roll a 3 on the next roll is approximately 16.67%.
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show that β=3α, by calculating the infinitesimal change in volume dv of a cube with sides of length l when the temperature changes by dt.
Answers
To show that β=3α, where β represents the volumetric thermal expansion coefficient and α represents the linear thermal expansion coefficient, we can calculate the infinitesimal change in volume (dv) of a cube with sides of length l when the temperature changes by dt.
The linear thermal expansion coefficient α is defined as the fractional change in length per unit change in temperature. Similarly, the volumetric thermal expansion coefficient β is defined as the fractional change in volume per unit change in temperature.
Let's consider a cube with sides of length l. The initial volume of the cube is \(V = l^3\). Now, when the temperature changes by dt, the sides of the cube will also change. Let dl be the infinitesimal change in length due to the temperature change.
The infinitesimal change in volume, dv, can be calculated using the formula for differential calculus:
\(\[dv = \frac{{\partial V}}{{\partial l}} dl = \frac{{dV}}{{dl}} dl\]\)
Since \(V = l^3,\) we can differentiate both sides of the equation with respect to l:
\(\[dV = 3l^2 dl\]\)
Substituting this back into the previous equation, we get:
\(\[dv = 3l^2 dl\]\)
Now, we can express dl in terms of dt using the linear thermal expansion coefficient α:
\(\[dl = \alpha l dt\]\)
Substituting this into the equation for dv, we have:
\(\[dv = 3l^2 \alpha l dt = 3\alpha l^3 dt\]\)
Comparing this with the definition of β (fractional change in volume per unit change in temperature), we find that:
\(\[\beta = \frac{{dv}}{{V dt}} = \frac{{3\alpha l^3 dt}}{{l^3 dt}} = 3\alpha\]\)
Therefore, we have shown that β = 3α, indicating that the volumetric thermal expansion coefficient is three times the linear thermal expansion coefficient for a cube.
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Rewrite 5 square root 3^4 as an expression with a rational exponent. Then mark this number on the graph of f (x) = 3^x. Explain how you know where to place it.
Answers
Step-by-step explanation:
the nth root of something can be expressed as 1/n as exponent.
e.g. sqrt(2) = 2^(1/2).
so, the 5th root of 3⁴ = 3^(4/5)
for f(x) = 3^x
that means that x = 4/5 = 0.8.
so, I go on the x-axis to the right to 0.8, and from there straight up to the actual curve.
that is the desired point.
can someone help me with this problem? Thank you!
Answers
Answer:
To find the area of a square, multiply the 2 sides by each other (all sides are equal)
S=14×14=196
196m^2
(3.12*10^15)-(4.26*10^14)
Answers
Answer:
2.694*10¹⁴Step-by-step explanation:
3.12*10¹⁵ - 4.26*10¹⁴ = 31.2*10¹⁴ - 4.26*10¹⁴ =(31.2 - 4.26)*10¹⁴ = 26.94*10¹⁴ = 2.694*10¹⁴Answer:
2.694*10¹⁴
hops this helps
Given (x – 7)2 = 36, select the values of x.
---x = 13
---x = 1
x = –29
x = 42
Given (x – 1)2 = 50, select the values of x.
x = –49
x = 51
---x=1+5 root 2
---x=1-5 root 2
Answers
For this first one it's:
A. ---x = 13
B. ---x = 1
Second one:
C. ---x=1+5 root 2
D. ---x=1-5 root 2
Abigail owns a food truck that sells tacos and burritos. She only has enough supplies
to make 130 tacos or burritos. She sells each taco for $3 and each burrito for $6.
Abigail must sell no less than $600 worth of tacos and burritos each day. If x
represents the number of tacos sold and y represents the number of burritos sold,
write and solve a system of inequalities graphically and determine one possible
solution.
Answers
Answer:
Inequality 1: x + y ≤ 1.50
Inequality 2: 4x + 6.5y ≥ 780
So, the possible solution can be x = 20, y = 1.20
(1) The inequalities can be written as:-
Inequality 1: x + y ≤ 1.50
Inequality 2: 4x + 6.5y ≥ 780
(2) The possible solution for the value of x and y can be x = 0.30, y = 1.20.
What is inequality?The inequality expressions are the mathematical equations related by each other by using the signs of greater than or less than. All the variables and numbers can be used to make the equation of inequality.
Given that Abigail owns a food truck that sells tacos and burritos. She only has enough supplies to make 130 tacos or burritos. She sells each taco for $3 and each burrito for $6. Abigail must sell no less than $600 worth of tacos and burritos each day.
The inequalities can be written as:-
Inequality 1: x + y ≤ 1.50
Inequality 2: 4x + 6.5y ≥ 780
The two inequalities can be solved as:-
x + y = 1.50 or x= 1.50 - y
4x + 6.5y = 780
4( 1.5 - y ) + 6.5y = 780
6 - 4y + 6.5y = 780
y = 1.20
The value of x will be,
x= 1.50 - y
x= 1.50 - 1.2
x = 0.30
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Help pleaseeeeeee!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:it’s the third answer choice
Step-by-step explanation:
Nancy's morning routine involves getting dressed, eating breakfast, making her bed, and driving to work. Nancy spends ⅓ of the total time in the morning getting dressed, 10 minutes eating breakfast, 5 minutes making her bed, and the remaining time driving to work. If Nancy spends 35 ½ minutes getting dressed, eating breakfast, and making her bed, how long is her drive to work?
Answers
Answer: 26 minutes
Step-by-step explanation:
Assume that the total time spent in the morning which includes getting dressed, eating breakfast, making her bed, and driving to work is x.
Time getting dressed = ¹/₃x
Eating breakfast = 10 mins
Making bed = 5 mins
¹/₃x + 10 + 5 = 35 ½
¹/₃x + 15 = 35 ½
¹/₃x = 35 ½ - 15
¹/₃x = 20 ½
x = 20 ½ / ¹/₃
= 61.5 minutes
Total morning routine is is 61.5 mins
Time taken to drive to work is;
= 61.5 - 35 ½
= 26 minutes
can someone help me with this
how do u get for 6x+8=2x-7 to 4x+8=-7
Answers
Answer:17
Step-by-step explanation:
Answer: Look for like terms and then leave the equation as it is :)
Step-by-step explanation: Hope this helps :)
Write a function g whose graph represents the indicated transformation of the graph of f. f(x)= | 4x+3| + 2 translated 2 units down is g (x)
Answers
Answer:
g(x) = |4x+3|
Step-by-step explanation:
g(x) = f(x) - 2
=|4x+3|
Write the equation of the line in fully simplified slope-intercept form.
Answers
hope this helps! :)
AYOOO PLZ HELP ASAP!!!
Answers
Answer:
B.
Step-by-step explanation:
Well we know that
\(224=2^{5} *7\)
so we can get the 2 outside of the radical
\(x^{11} =(x^{5} )^{2} *x\)
and we can get the x^2 outside too.
\(y^8=y^5*y^3\)
and we also can get y outside.
so we have:
\(2x^{2}y\sqrt[5]{7xy^3}\)
Someone pls help?????????????????????????
Answers
Answer:
p=m*9+9
So 45/4=9
9*4=36
36+9=45
Check with table
5*9=45
45+9=54
6+9=54
63+9=63
7+9=63
63+9=72
So as you can se the equation is p=9*m+9
Hopes this helps!
The dimensions of a fish tank are 5 ft by 4 ft by 2 ft. There are 40 fish in the tank. What is the population density of the fish tank?
A. 2.0 fish per cu. ft.
B. 1.0 fish per cu. ft.
C. 0.5 fish per cu. ft.
Answers
When solving this system of equations by elimination, which could be the resulting equation when the variable y has been eliminated?
9x + 3y = 12
2x + y = 5
answers:
A) 3x = -3
B) 11x = 17
C) 15x = 27
D) 7x = 7
Answers
When solving this system of equations by elimination, the resulting equation when the variable y has been eliminated is, 3x = -3.
Hence the correct option is (A).
The given system of linear equations is
9x + 3y = 12 ............... (i)
2x + y = 5 ................. (ii)
We have to solve the system using elimination method by eliminating the variable y first.
Coefficients of y in both equations are 3 and 1 respectively.
So, LCM(3, 1) = 3.
Multiplying the equation (ii) by 3 we get,
3(2x + y) = 3*5
6x + 3y = 15 ............. (iii)
Subtracting the equation (i) from the equation (iii) we get,
6x + 3y - 9x - 3y = 15 - 12
-3x = 3
3x = -3
so the resulting equation when the variable y has been eliminated is, 3x = -3.
Hence the correct option is (A).
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5. Janet Gilbert is director of labs. She has some extra capacity and has contracted with some small neighboring hospitals to run some of their lab tests. She has recently had a study conducted and has determined that her costs of these contracts are $10,000 of which $7,000 are for supplies and items related to each test. She currently charges an average of $10.00 per lab test. She is thinking of lowering her price by 20 percent in hopes of raising her current volume of 10,000 tests by 15 percent.
Answers
Answer:
Check the explanation
Step-by-step explanation:
Present predicted
Revenue - per test 10000 8000
Variable cost - per test 7000 7000
Contribution – per test 3000 1000
Net income (10,000 test – 11,500 tests) 30,000,000 11,500,000
Anya has $25,000 which she recently received from a trust fund, which she intends to invest in an account earning 12% annually. a) How many years would it take Anya to accumulate $40,000. b) If Anya's goal is to save $40,000 in just 3 years, what rate of return must she earn annually on her account. Show all workings and formulae
Answers
a) It would take Anya approximately 4 years to accumulate $40,000 with an annual interest rate of 12%. b) Anya must earn an annual rate of return of approximately 12.6% to save $40,000 in 3 years.
a) To calculate the number of years it would take Anya to accumulate $40,000, we can use the future value formula for compound interest:
Future Value = Present Value * (1 + interest rate)ⁿ
Where:
Future Value = $40,000
Present Value = $25,000
Interest rate = 12% = 0.12
n = number of years
Substituting the given values into the formula, we have:
$40,000 = $25,000 * (1 + 0.12)ⁿ
Dividing both sides of the equation by $25,000, we get:
(1 + 0.12)ⁿ = 40,000 / 25,000
(1.12)ⁿ = 1.6
To solve for n, we can take the logarithm of both sides of the equation:
n * log(1.12) = log(1.6)
Using a calculator, we find that log(1.12) ≈ 0.0492 and log(1.6) ≈ 0.2041. Therefore:
n * 0.0492 = 0.2041
n = 0.2041 / 0.0492 ≈ 4.15
b) To calculate the required rate of return for Anya to save $40,000 in just 3 years, we can rearrange the future value formula:
Future Value = Present Value * (1 + interest rate)ⁿ
$40,000 = $25,000 * (1 + interest rate)³
Dividing both sides of the equation by $25,000, we have:
(1 + interest rate)³ = 40,000 / 25,000
(1 + interest rate)³ = 1.6
Taking the cube root of both sides of the equation:
1 + interest rate = ∛1.6
Subtracting 1 from both sides, we get:
interest rate = ∛1.6 - 1
Using a calculator, we find that ∛1.6 ≈ 1.126. Therefore:
interest rate = 1.126 - 1 ≈ 0.126
To express the interest rate as a percentage, we multiply by 100:
interest rate = 0.126 * 100 = 12.6%
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Directions: Find the value of x.
Round each answer to the nearest tenth.
Answers
Answer:
this answer is not available at the moment
Step-by-step explanation:
5. On average, John eats 200 grams of sugar per
day. 85 grams is approximately equivalent to 3
ounces. How many ounces is this? 7.4E
A. 2 ounces
C. 3 ounces
B. 6 ounces
D. 7 ounces
Answers
Work Shown:
(85 grams)/(3 ounces) = (200 grams)/(x ounces)
85/3 = 200/x
85x = 200*3
85x = 600
x = 600/85
x = 7.0588 approximately
This rounds to 7 ounces when rounding to the nearest whole number.