Answer:
The answer should be 8.
Step-by-step explanation:
the output voltage for an electric circuit is specified to be 130.asampleof40independentreadings on the voltage for this circuit gave a sample mean 128.6 and standard deviation 2.1.test the hypothesis that the average output voltage is 130against the alternative that it is lessthan 130.useatestwithlevel.05
Consequently, we deduce that the typical output voltage is under 130.
What is standard deviation ?In statistics, the standard deviation measures how widely distributed or how much a bunch of values can vary. While a big standard deviation shows that values are scattered throughout a wider range, a low standard deviation suggests that values are frequently close to the set's mean.
Here,
A defined output voltage of 130 is offered for an electrical circuit.
The voltage for this circuit was measured 40 times independently, with a sample mean of 128.6 and a standard deviation of 2.1.
Let's say the average output voltage.
Null Hypothesis, [tex]H_{0}[/tex]: = 130 denotes that the average output voltage is 130.
A Different Hypothesis, [tex]H_{A}[/tex] : < 130 ( it indicates that the average output voltage is below 130.)
One-sample t test statistics will be employed in this case because we are unsure about the population standard deviation;
T.S. = ~ [tex]\frac{X^{'}-u }{\frac{s}{\sqrt{n} } }[/tex]
In this case, sample mean output voltage equals 128.6
Sample standard deviation equals 2.1, or s.
n = 40 for the sample of independent readers.
Thus, test statistics equal.
So, test statistics = ~[tex]\frac{128.6-130 }{\frac{2.1}{\sqrt{40} } }[/tex]
= -4.216
Now, for a left-tailed test with a 0.05 level of significance, the t table provides a critical value of -1.685 at 39 degrees of freedom. We have enough data to reject our null hypothesis because it will fall in the rejection region because our test statistics are smaller than the crucial value of t, which is -4.216 -1.685.
Consequently, we deduce that the typical output voltage is under 130.
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Drag the tiles to the correct boxes to complete the pairs.
Match the one-to-one functions with their inverse functions.
f(x)=-17
f(x)=x-10
Inverse Function
f-¹(x) = 5x
ƒ-¹ (x) = ²
f-¹ (z) = x + 10
f-¹ (z) = 3(z+17)
f(x) = √2x
Function
f(x) = //
The inverse of a function is f^-1(x) = 3(x + 17)/2 and f^-1(x) = x + 10
What is function ?
An expression, rule, or law in mathematics that specifies the relationship between an independent variable and a dependent variable (the dependent variable).
An example of a rule is a function, which produces one output for a single input. Alex Federspiel is the picture's author. Y=X2 is an illustration of this. You only get one output for y if you enter anything for x. The fact that x is the input value leads us to say that y is a function of x.
Functions can be roughly categorized into four sorts. based on the element Numerous to one function, onto function, one to one and onto function, into function.
1.f(x) = 2x/3 - 17
Finding the inverse
y = f(x)
= 2x / 3 - 17
Exchange the variables, x for y and y for x
x = 2y / 3 - 17
Isolate the variable y
Adds 17 both sides
x + 17 = 2y / 3
Multiply by 3 both sides
3(x+17) = 2y
Divide by 2 both sides
y = 3(x + 17)/2
let f^-1(x) = y
f^-1(x) = 3(x + 17)/2
2. f(x) = x - 10
Finding the inverse
y = f(x)
= x - 10
Exchange the variables, x for y and y for x
x = y - 10
Isolate the variable y
Adds 10 both sides
y = x + 10
Multiply by 3 both sides
3(x+17) = 2y
let f^-1(x) = y
f^-1(x) = x + 10
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Triangle PQR is similar to triangle STU. Find the measure of side ST. Figures are not
drawn to scale.
Answer:
ST = 2.6
Step-by-step explanation:
since the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{ST}{PQ}[/tex] = [tex]\frac{SU}{PR}[/tex] ( substitute values )
[tex]\frac{ST}{18}[/tex] = [tex]\frac{5.2}{36}[/tex] ( cross- multiply )
36 × ST = 18 × 5.2 = 93.6 ( divide both sides by 36 )
ST = 2.6
the probability that a restaurant passes its health department inspection is 0.94. the owners of 58 restaurants are surveyed to see if their restaurants passed inspection. For each binomial experiment described, find the mean, the variance and the standard deviation!
Mean = 54.52
V(X) = 3.2712
S.D = 1.8086
To calculate the mean (expected value) of a binomial distribution B(n, p) you need to multiply the number of trials n by the probability of successes p , that is: mean = n × p
Where, n= 58
p = 0.94
Therefore, mean = 54.52
To find the Variance of the given binomial distribution
Variance = np(1-p)
=58 X 0.94 ( 1- 0.94)
= 54.52 X 0.06
V(X) = 3.2712, when we know the variance and the mean then we can calculate the standard deviation easily
To find the standard deviation of the given binomial experiment = [tex]\sqrt{np(1-p)}[/tex]
= [tex]\sqrt{3.2712}[/tex]
S.D = 1.8086
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a = 1212(b1 b2)h find the area of the trapezoid.
The area of trapezoid with sides 125ft and 81ft with height 75ft is: 7,725ft².
What is the area of trapezoid?A trapezoid is a quadrilateral that is convex. There are at least two parallel sides to a trapezoid. The parallel sides are referred to as bases, while the non-parallel sides are referred to as legs.The bases are the two parallel sides, and height is always the perpendicular distance from one base to the opposite. The area of this parallelogram is equal to half of its height (half of the trapezoid's height) times its base (sum of the trapezoid's bases)Area of a trapezoid,
A = 1/2 x (b1 x b2)h
Here given,
b1 = 125ft
b2 = 81ft
h = 75ft
By substituting the values ,
A = 1/2 ( 125 + 81 ) x 75
A = 1/2 x 206 x 75
A = 0.5 x 206 x 75
A = 7,725 ft²
The complete question is "To answer the question, use the formula for the area of a trapezoid A=h((b1+b2)/2), where A is the area, b1 and b2 are the lengths of the bases, and h is the height. What is the area of grass on a trapezoidal field with a height of 75 feet and bases of 125 feet and 81 feet? HINT: Use the values given for h, b1, and b2.".
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The area of trapezoid is 7725 ft²
What is a trapezoid?
In American and Canadian English, a quadrilateral with at least one pair of parallel sides is referred to as a trapezoid. It is referred to as a trapezium in British and other varieties of English.
In Euclidean geometry, a trapezoid is a convex quadrilateral by definition. The bases of the trapezoid are the parallel sides. If the remaining two sides are parallel, they are referred to as the legs (or lateral sides); if not, the trapezoid is a parallelogram, and there are two pairs of bases. Unlike the specific situations below, a scalene trapezoid is a trapezoid that lacks equal-sized sides.
Given, the expression for the area a is given as
a = [(b₁ + b₂)/2]*h
where b₁ = 125ft, b₂ = 81ft, h = 75ft.
Substituting these values in the above area expression, we get,
a = [(125+81)/2]*75
a = 103*75
a = 7725 ft²
Thus, the area of trapezoid is 7725 ft²
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which is the graph of y=2(x-3)^2 2
what is the missing side length in this right triangle?round the answer to the nearest tenth.5.7 cm5.7 cm12.8 cm12.8 cm23.6 cm
The missing side length in the given right triangle is d) 23.6cm if the other two sides lengths are 14cm and 19cm respectively. So, correct option is d.
We know very well that if a triangle is a right angled it means that Pythagorean Theorem is applicable on it.
Pythagorean Theorem states that if a triangle is right angled then the sum of square of two sides which are originating from that angle which is 90 degree is equal to square of third side.
In other words,a²+b²=c² where a and b are two sides originating from 90 degree angle and c is the third side which is opposite of the 90 degree angle.
Now, we have a=14 and b=19cm,we need to find value of c.
So, putting the values, we get
=>14² + 19² =c²
=>c²=361+196
=>c² = 557
=>c=√557
=>c=23.6cm
Hence, missing side length is d)23.6cm
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(Complete question) is:
what is the missing side length in this right triangle if the other two sides length are 14cm and 19cm respectively ? round the answer to the nearest tenth.5.7 cm5.7 cm12.8 cm12.8 cm23.6 cm
3. Write the equation of a line that is perpendicular to y =V3 x + 3 and goes through the point (-1, -2).
Step-by-step explanation:
is that root 3 x..........?
The Oakland Tigers are having team shirts made. One option is to pay Susan's Tees a $34 setup fee and then buy the shirts for $10 each. Another option is to go to City Printing, paying $23 for a setup fee and an additional $11 per shirt. The team parent in charge of the project notices that, with a certain number of shirts, the two options have the same cost. What is the cost? How many shirts is that?
To find the cost of the shirts, we need to set up a system of equations. Let x be the number of shirts.
Option 1: 34 + 10x = Option 2: 23 + 11x
We can solve this system of equations by subtracting the equations:
34 + 10x - 23 - 11x = 0
11x - 23 = 0
11x = 23
x = 23 / 11
Therefore, the cost is the same when the team orders 23/11 = 2 shirts.
Complete the coordinate proof of the theorem.
Given: ABCD is a square.
Prove: The diagonals of ABCD are perpendicular.
The coordinates of the square be
A (0, 0)
B (a, 0)
C (a, a)
D (0, a)
slope of line AC = 1
How to find the slope of ACSlope is the rate of change of the line AC this is solved by taking the ratio of change in y to change in x
A (0, 0) and C (a, a)
= (0 - a) / (0 - a)
= -a / -a
= 1
Slope of line BD = -1
the product of the slopes
= 1 * -1
= -1
therefore the diagonals of the ABCD are perpendicular since the product of the slope is -1
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suppose steph curry is fouled while shooting a three-point goal and misses his shot. the probability of steph curry making a free throw is 90%. what is the probability that he makes the first and second free throw, but misses the third free throw?
The probability that he makes the first and second free throw, but misses the third free throw is 7.7 %.
Given :
suppose steph curry is fouled while shooting a three-point goal and misses his shot. the probability of steph curry making a free throw is 90%.
probability :
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.
P ( makes 2 then misses 1 ) = P ( make 1st ) and P ( make 2nd ) and P ( miss 3rd )
= 0.906 *0.906 * 0.094
= 0.077
= 7.7 %
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Vickie made a recipe for 144 ounce of cented candle wax how many 1 cup candle mold can he fill with the recipe
The number of 1 cup candle mold which the can fill with 144 ounce of cented candle wax is 18.
As we know ,
1-cup= 8 ounces
On dividing ,
144 by 8 = 18
Therefore , the no of cups which can be filled is 18 .
One of the fundamental arithmetic operations in mathematics is the division, which is the process of dividing a group of items into equal pieces. Every day, we might employ the division approach in various situations.
Any number divided by 0 has no specified result. The outcome in this situation can be expressed as infinite. For instance, 15/0 =, which is undefined.
Dividend :
The division procedure shown below is used to divide any expression.
Dividend is equal to Divisor Quotient + Remainder.
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Which of the following is a criteria for construction of triangles A AAA B SAS C ss D None?
A criteria for construction of triangle include the following: B. SAS.
What is a triangle?In Mathematics, a triangle can be defined as a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.
Generally speaking, the side, angle, side (SAS) criterion is essential for the construction of any triangle such as the following:
Isosceles triangleObtuse triangleEquilateral triangleScalene triangleRight-angled triangleBased on the side, angle, side (SAS) criterion, a statement which is necessary to prove that two (2) triangles are congruent is that one of its side must be equal to another side because their angles are equal in measure or magnitude.
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Assuming the cost of an associate leaving within 90 day is $3,000 what will be your facility's approximate cost of early turnover for this year?
The facility's approximate cost of early turnover for this year is $45000.
The cost of turnover is the cost associated with turning over one position. This calculation includes the cost of hiring for that position, training the new employee, any severance or bonus packages, and managing the role when it is not filled. Every company will experience some turnover.
Turnover cost refers to expense both tangible or intangible associated with replacing an employee. It is extremely high: it's estimated that losing an employee can cost a company 1.5-2 times the employee's salary.
Shift A has number of associated leaving the facility within 90 days is 8, per associated cost is $3000 and total cost is $24000.
Shift B has number of associated leaving the facility within 90 days is 5, per associated cost is $3000 and total cost is $15000.
Shift C has number of associated leaving the facility within 90 days is 2, per associated cost is $3000 and total cost is $6000.
Shift D has number of associated leaving the facility within 90 days is 0, per associated cost is $3000 and total cost is $30.
Therefore
Total cost = $24000 + $3000 + $6000
Total cost = $45000
Hence the facility's approximate cost of early turnover for this year is $45000.
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Make X the
subject of m=n+x/p
Answer:
x = pm - pn
Step-by-step explanation:
m = n + [tex]\frac{x}{p}[/tex] Subtract n from both sides
m - n = n - n + [tex]\frac{x}{p}[/tex]
m-n = [tex]\frac{x}{p}[/tex] Multiply both sides by p
p(m - n) = ([tex]\frac{p}{1}[/tex]) [tex]\frac{x}{p}[/tex] [tex]\frac{p}{1}[/tex] means the same as p
pm - pn = x Distribute the p
x = pm - pn
HELP PLEASE! ILL MARK BRAINILEST.
Answer:
Below
Step-by-step explanation:
Area of a triangle = 1/2 b * h
Triangle A = 1/2 (4 * 10) = 20 units^2 ( I got the '10' from answers below)
Triangle B must be b = 2 h = 2 (for area = 2)
Rectangle C must be 2 x 2 ( for area = 4)
Triangle D = 1/2 (2+2+6)*2 = 10 units ^2
Total = 20 + 2+4+ 10 = 36 units^2
Answer:
32
Step-by-step explanation:
Area of a triangle = 1/2bh
b = base
h = height
Area of a square = bh, where b=h
A: 1/2(2+2+6)(4) = 20
B: 1/2(2)(2) = 2
C: (2)(2) = 4
D: 1/2(6)(2) = 6
Total = A+B+C+D = 20+2+4+6 = 32
**the way the dimension "6" is labeled is a little confusing. I assumed 6 is the base of triangle D
How do you write 145,000,000,000 in scientific notation?
The scientific notation is [tex]1.45*10^{11}[/tex]
Scientific notation is a way to present extremely large or extremely small numbers in a more understandable way. We are aware that full numbers can go on forever, but we are unable to write such enormous figures on paper. Additionally, a simpler method of representation was required for the numbers that appear at the millions place after the decimal. This makes it challenging to represent a small number of integers in their enlarged form. We, therefore, employ scientific notations.
We need to move the decimal point 11 places to the left therefore the exponent for the 10s term will be positive:
[tex]145,000,000,000=145,000,000,000.0=1.45*10^{11}[/tex]
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Determine the coordinates of W(−7, 4) after a reflection in the line y=9
The coordinate of the reflection about the line y = 9 is thus (7, 4)
What is reflection?Reflection is a transformation in which an object is flipped over a line.
How to find the coordinates of W(−7, 4) after a reflection in the line y = 9?We know that to reflect W(-7, 4) over the line y = 9, the distance of the point W and its reflection from the line y = 9 must be the same.
The distance of point W(-7, 4) from the line y = 9 is d = 9 - (- 7) = 9 + 7 = 15
So, the x - coordinate of the reflection from the line y = 9 is x = 15 - 9 = 7
Since the y - coordinate of the point W, does not change, the coordinate of the reflection about the line y = 9 is thus (7, 4)
So, the coordinate of the reflection about the line y = 9 is thus (7, 4)
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How do you simplify the expression 1+tan^2θ?
The tangent theta is a Trigonometric function, the simplified value of 1 + tan²θ is sec²θ.
What is Trigonometry function ?Trigonometric functions are also called circular functions. They are easily defined as functions of triangle angles. This means that the relationships between the angles and sides of triangles are given by these trigonometric functions. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. Also read the triangle identity here.
We have given an expression, 1 + tan²θ
we know that , tanθ = sinθ /cosθ
sin²θ + cos²θ =1 and 1/cosθ = secθ
Now, 1 + tan²θ = 1 + sin²θ/cos²θ
taking LCM cos²θ
=> 1 + tan²θ = (cos²θ + sin²θ )/cos²θ
=> 1 + tan²θ = 1/cos²θ
=> 1 + tan²θ = sec²θ
Hence, the required expression is sec²θ.
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A container Can hold a maximum of 17.5 pounds. There are two objects in the container. The first object weighs 2.65 pounds and the second object weighs 10.8 pounds . How many more pounds can container hold
Answer:
4.05 pounds
Step-by-step explanation:
You want to know the weight that can be added to 2.65 pounds and 10.8 pounds if the total is to be 17.5 pounds.
Additional weightThe available capacity is the difference between the capacity and the weight already in the container:
17.5 lb -2.65 lb -10.8 lb = 4.05 lb
The container can hold up to 4.05 more pounds.
Jack and Diane are jogging back and forth along a one-mile path. They started out at 9:00 A.M. from opposite ends of the path. They passed each other in 10 minutes when Diane has gone 1/3 mile. What time will they meet at the same end of the path? You have to assume they keep jogging at the same speeds.
Explain:
If they keep running at the same pace, they will eventually meet at the same end of the course after 30 minutes.
what is speed ?The rate at which distance and time change is what is meant by speed. It has the aspect of temporal and spatial distance. The basic unit of distance with the basic unit of time are combined to form the SI unit of speed. This means that the SI unit of speed is the metre per second.
The way that the issue is described is vague.
I believe what is being said here is that they continue to jog at the same speed.
Half a mile per ten minutes is Diane's speed.
2/3 miles per 10 minutes is Jack's speed.
The numerator and denominator (top and bottom of the ratio) must be multiplied by the factor between 10 minutes and an hour (60 minutes) in order to convert this to a standard miles per hour format.
60/10 = 6.
In order to convert the above speeds to miles per hour, we must multiply both speeds by 6.
Diane: (1/3 divided by 6)/hour equals 2 miles/hour
Jack: (4 miles per hour) = (2/3 x 6) / hour
Diane will complete one length in the time it takes Jack to complete a round trip (back) because he runs twice as fast as she does.
Running a mile at a speed of two miles per hour takes Diane 30 minutes.
Running 2 miles (back and forth) at a speed of 4 miles per hour will take Jack 30 minutes.
So after 30 minutes they will meet again at his starting point.
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a street light is at the top of a 16 foot tall pole. a 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. how fast is the tip of her shadow moving when she is 35 feet from the base of the pole?
By using property of similar triangle and speed, the answer is 4.2ft/sec
What is a triangle?A polygon with three edges and three vertices is called a triangle. It is one of the fundamental geometric shapes.
What is mean by similar triangle?Similar triangles always have comparable sides that are in the same ratio.
We can compare the ratio of sides of similar triangle for the calculations.
x= shadow length at 35feet away.
Using the property of Similar Triangle we get,
[tex]\frac {16}{35+x} = \frac {6}{x}[/tex]
x=21 feet
Velocity of shadow= 21/5
=4.2 feet/sec
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i need help answering with steps
-40-2(3m+1/2) =7m-2
Answer:
Step-by-step explanation:
-Start by distributing the 2, -2(3m+1/2).
-Getting -6m-1, your equation should look like -40-6m-1=7m-2
-Combine -40 and -1 to get -41, and then add the -6m to the other side.
-The equation should look like -41=13m-2. Add the 2 to the -41 to get -39.
-Now, we have -39=13m. divide both sides by 13 to get m=-3.
if a snowball melts so that its surface area decreases at a rate of 1cm2/min1cm2/min, find the rate at which the diameter decreases when the diameter is 10cm10cm.
The rate at which the diameter decreases is -0.16cm/min
Diameter:
In geometry, diameter refers the straight line that goes from one side to the other side of a circle, passing through the center.
Given,
Here we need to find if a snowball melts so that its surface area decreases at a rate of 1cm^2/min, find the rate at which the diameter decreases when the diameter is 10 cm. Finish solving the problem. Write an equation that relates the quantities.
Here we obtained the equation that relates the surface area S and the diameter D of a sphere given by
=> S = πD²
Now, we have differentiate the equation as,
=> dS/dt = d/dt (πD²)
=> dS/dt = π d/dt (D²)
=> dS/dt = 2πD dD/dt
Now, Solving for dD/dt, which is the unknown for this problem, we have
=> dD/dt = 1/2πD x dS/dt
Now, substituting the given values, the rate of change of the diameter when D=10 cm and dS/dt as (-1) is
=> dD/dt = 1/2 x π x 10 (-1)
=> dD/dt = -1/20π
=> dD/dt = -0.16cm/min
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what number would you be rather guessing correctly? the number of rolls of a fair six-sided die will
The number of rolls of a fair six sided die will be 1/6.
A cube (plural "cubes") is a solid with marks on each face. Since the faces are usually all of the same shape, the dual of the Platonic solids and the Archimedean solids is a natural choice. You can "roll" a die by throwing it in the air and letting it rest on one of its faces. Dice are used in many games to choose random numbers to bet on, and are used in board and role-playing games to determine how many squares to move, the outcome of a competition, etc. Coins are a degenerate two-sided case of dice.
In probability theory, an outcome is the possible outcome of an experiment or trial. Each possible outcome of a given experiment is unique, and different outcomes are mutually exclusive (only one outcome occurs in each trial of the experiment). All possible outcomes of an experiment form an element of sample space.
According to the question,
When a dice is thrown, number of possible outcomes =
1, 2, 3, 4, 5, 6 = 6 = n
Here, 6 occurs one time, i.e. m = 1
Thus, Probability of getting a number 6 = 1/6
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A Blue spruce grows an average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 48 inches tall, and a Hemlock is 70 inches tall, when would you expect the trees to be the same height? What height will that be? Write and solve the system of equations for this problem.
In 12 years the two trees will be at the same height of 120 inches for the given condition of the equation.
What is an equation?
A mathematical statement called an equation demonstrates the equality of two mathematical expressions.
Based on the given condition of a question,
Let X be the required time duration, and then the equation will be
or, (48+6X)=(72+4X)
or, 48+6X=72+4X
or, 6X-4X=72-48
or, 2X=24
or, X=24/2
or, X=12 year
And height = 48 + 6*12 = 48 + 72 = 120 inches
Hence, In 12 years the two trees will be at the same height of 120 inches.
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When a number is increased by 6, the answer is 13. Translate into linear equation but do not solve.
Answer:
n + 6 = 13
Step-by-step explanation:
let n be the number, then increasing by 6 means adding 6 to it , so
n + 6 = 13 ← linear equation
a city ordinance requires that more than 75% of its residents must agree to the construction of new public buildings (using tax dollars) before any such structures can be built. a proposal has been made to build a new recreational facility in the city, and sponsors of the proposal want to conduct a small survey to see if it would be approved if put to an official vote of all residents. a simple random sample of 150 residents revealed that 123 supported a change (and 27 did not). what is the value of the standardized test statistic for this significance test?
The value of the standardized test statistic for this significance test is 1.7265
Standardized test statistic
In statistics, the standardized test statistic means a way for you to compare your results to a “normal” population.
Given,
A city ordinance requires that more than 75% of its residents must agree to the construction of new public buildings (using tax dollars) before any such structures can be built. a proposal has been made to build a new recreational facility in the city, and sponsors of the proposal want to conduct a small survey to see if it would be approved if put to an official vote of all residents. a simple random sample of 150 residents revealed that 123 supported a change (and 27 did not).
Here we need to find the value of the standardized test statistic for this significance test.
While we looking into the given question we have identified the following values,
Random samples = 150
Supported values = 123
Not supported = 23
Aggreged percentage = 75%
So, the null and alternative hypotheses were as follows: H0: p = 0.75 and Ha: p ≠ 0.75.
Then the value of Standardized test statistic is calculated as,
=> (150 - 123) / (75 / √23)
=> 1.7265
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One-third of the seventh garde class bought tickets for the seventh grade dance. Then, 32 students bought tickets at the door. If there were 158 students at the dance, how many total students are there in seventh grade?
There are total 378 students in seventh grade.
What is arithmetic?
Arithmetic is an elementary a part of arithmetic that consists of the study of the properties of the standard operations on numbers—addition, subtraction, multiplication, division, operation, and extraction of roots.
Main body:
The total number of students that are in the seventh grade are 378 students.
The number of students that had already bought tickets will be:
= 158 - 32 = 126 students.
Therefore, based on the information given, the equation to calculate the total students will be:
= (1/3 × x) = 126
x = 3 × 126
x = 378
Therefore, there are 378 students.
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What is fg(x) given that f(x) equals x² + 7x and g(x) equals -2x² + 3x-5
Answer:
[tex] \huge{ \boxed{f(g(x)) = 4 {x}^{4} - 12 {x}^{3} + 15 {x}^{2} - 9x - 10 \\ }}[/tex]
Step-by-step explanation:
[tex]f(x) = {x}^{2} + 7x \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ g(x) = - 2 {x}^{2} + 3x - 5[/tex]
To find f(g(x)), substitute g(x) into f(x). That is for every x found in f(x) replace it with g(x) and solve.
[tex]f(g(x)) = {( - 2 {x}^{2} + 3x - 5) }^{2} + 7( - {2x}^{2} + 3x - 5) \: \: \: \: \: \: \: \: \: \: \\ \\ = (4 x^4 - 12 x^3 + 29 x^2 - 30 x + 25) + (-14 x^2 + 21 x - 35) \\ = 4 {x}^{4} - 12 {x}^{3} + 29 {x}^{2} - 14 {x}^{2} - 30x + 21x + 25 - 35 \\ \\ = 4 {x}^{4} - 12 {x}^{3} + 15 {x}^{2} - 9x - 10[/tex]
We have the final answer as
[tex]f(g(x)) = 4 {x}^{4} - 12 {x}^{3} + 15 {x}^{2} - 9x - 10 \\ [/tex]
