Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers.f(x)=sqrt(x)+2g(x)=x^2+7f(g(x))= ?g(f(x))= ?
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Answer:
[tex]\begin{gathered} \begin{equation*} f(g(x))=\sqrt{x^2+7}+2 \end{equation*} \\ \begin{equation*} g(f(x))=x+4\sqrt{x}+11 \end{equation*} \end{gathered}[/tex]Explanation:
Given the functions f(x) and g(x) below:
[tex]\begin{gathered} f(x)=\sqrt{x}+2 \\ g\mleft(x\mright)=x^2+7 \end{gathered}[/tex]Part A
We want to find the simplified form of f(g(x)).
[tex]f(x)=\sqrt{x}+2[/tex]Replace x with g(x):
[tex]f(g(x))=\sqrt{g(x)}+2[/tex]Finally, enter the expression for g(x) and simplify if possible:
[tex]\implies f\mleft(g\mleft(x\mright)\mright)=\sqrt{x^2+7}+2[/tex]Part B
We want to find the simplified form of g(f(x)). To do this, begin with g(x):
[tex]g\mleft(x\mright)=x^2+7[/tex]Replace x with f(x):
[tex]g(f(x))=[f(x)]^2+7[/tex]Finally, enter the expression for f(x) and simplify if possible:
[tex]\begin{gathered} g\mleft(f\mleft(x\mright)\mright)=(\sqrt{x}+2)^2+7 \\ =(\sqrt{x}+2)(\sqrt{x}+2)+7 \\ =x+2\sqrt{x}+2\sqrt{x}+4+7 \\ \implies g(f(x))=x+4\sqrt{x}+11 \end{gathered}[/tex]Therefore:
[tex]\begin{equation*} g(f(x))=x+4\sqrt{x}+11 \end{equation*}[/tex]Related Questions
write 0.751 as a percentage
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To convert decimal numbers to percentage, what we need to do is to multiply the decimal number by 100, and we will get the representation as a percentage.
In this case we have the decimal number:
[tex]0.751[/tex]We multiply that number by 100 to write is as a percentage:
[tex]0.751\times100=71.5[/tex]Answer: 75.1%
I need help on a problem
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As shown in the figure:
AB || CD
AD || CB
We need to prove AB = CD
So, the proof will be as follows:
Statements Reasons
0. AB || CD Given
,1. m∠BAC = m∠DCA Alternate angles are congruent
,2. AD || CB Given
,3. m∠BCA = m∠DAC Alternate angles are congruent
,4. AC = CA Reflexive property
,5. ΔBAC ≅ ΔDCA By A.S.A [angle-side-angle] postulate
,6. AB ≅ CD CPCTC
kName:ID:06/22/1973Time Remaining:00:55:49Teresa KundrataA car costs $14,000. The loan company hasasked for 1/10 of the cost of the car as adown payment what the down payment
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Given:
The cost of the car is $14,000.
Then 1/10 of the cost of the car is
[tex]\frac{1}{10}\times14000=1400[/tex]Hence, the down payment is $1400.
You are scuba diving at 120 feet below sea level. You begin to ascend at a rate of 4 feet per second.a. Where will you be 10 seconds after you begin your ascension? b. How long will it take to reach the surface?
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The ascension can be modeled using the function:
[tex]d(t)=d_0-r\cdot t[/tex]Where d is the number of feet below the sea level at time t (in seconds), d₀ is the initial "depth", and r is the ascension rate.
From the problem, we identify:
[tex]\begin{gathered} r=4\text{ feet per second} \\ d_0=120\text{ feet} \end{gathered}[/tex]Then:
[tex]d(t)=120-4t[/tex]a)
After 10 seconds, we have t = 10:
[tex]\begin{gathered} d(10)=120-4\cdot10=120-40 \\ \\ \Rightarrow d(10)=80\text{ feet} \end{gathered}[/tex]After 10 seconds, we will be 80 feet below sea level.
b)
To find how long will it take to reach the surface, we need to solve the equation d(t) = 0.
[tex]\begin{gathered} d(t)=0 \\ 120-4t=0 \\ 4t=120 \\ \\ \therefore t=30\text{ seconds} \end{gathered}[/tex]We will reach the surface after 30 seconds.
Debra is playing a role-playing game with her friends. She will roll dice to determine if her character unlocks a treasure chest. The probability of her unlocking the treasure chest is 3/10. Find the odds in favor of her character unlocking the treasure chest.
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Probability of Debra unlocking the treasure chest, P(unlocking) = 3/10
Probability of Debra not unlocking the treasure chest,
P( not unlocking) = 1 - 3/10
P( not unlocking) = 7/10
[tex]undefined[/tex]Write the equation of the line (in standard form) that goes through point (5,-1) and is parallel to the equation 3x + 2y =19.
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The equation of the line that goes through the point (5, -1) and is parallel to the given equation is; 3x + 2y = -7.
What is the equation of the line that gOES through the point given and is parallel tomthe given equation?Recall from line geometry that parallel lines have equal slopes.
Therefore, by computing the slope of the equation given as follows; we have;
3x + 2y = 19.
By rearranging the equation in the slope-intercept form; y = mx + c; we have;
y = (-3/2)x + 19/2
Therefore, the slope of both lines is same and is; -3/2.
Therefore, By using the point slope equation of a straight line; (y - y1) = m (x - x1); we have;
(y - (-1)) = -3/2(x - 5)
2y + 2 = -3x - 5
3x + 2y = -7.
Ultimately, the required equation is; 3x + 2y = -7.
Read more on slope of parallel lines;
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need help with part a with a summary and all work shown to help me understand better
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ANSWER:
[tex]\left(16u^{\frac{1}{3}}\right)^{\frac{3}{4}}=8\sqrt[4]{u}[/tex]STEP-BY-STEP EXPLANATION:
We have the following expression:
[tex]\left(16u^{\frac{1}{3}}\right)^{\frac{3}{4}}[/tex]When you raise an exponent to another exponent, multiply therefore:
10 ft to 8 ft The percent of change is
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The percent of change is computed as follows:
[tex]\text{percent of change = }\frac{new\text{ value }-previous\text{ value}}{previous\text{ value}}\cdot100[/tex]Substituting with data:
[tex]\begin{gathered} \text{ percent of change = }\frac{8-10}{10}\cdot100 \\ \text{ percent of change =}-20\text{ \%} \end{gathered}[/tex]Complete the proof that the point (-2, V5 ) does or does not lle on the circle centered at the origin and containing the point (0,3). Part 1 out of 4 The radius of the circle is
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We will have the following:
*First: We have that the equation of the circle will be given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]Here (h, k) is the coordinate of the center of the circle and r is the radius of the circle.
*Second: We will replace the center of the circle and determine the radius:
[tex]x^2+y^2=r^2[/tex]*Third: We determine the radius of the circle by using the point given:
[tex](0)^2+(3)^2=r^2\Rightarrow r^2=9\Rightarrow r=3[/tex]*Fourth: We have the following expression representing the circle:
[tex]x^2+y^2=9[/tex]So, we replace the point (-2, sqrt(5)) to determine whether or not it belongs to the circle, that is:
[tex](-2)^2+(\sqrt[]{5})^2=9\Rightarrow4+5=9\Rightarrow9=9[/tex]Thus proving that the point (-2, sqrt(5)) does lie in the circle.
Use one or more transformations to transform the pre-image (purple) onto the image (white). helppp
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The transformation required to transform the preimage in purple to the image in white is
Rotation 180 degreesTranslation to the right 14 unitsTranslation down 4 unitsWhat is transformation?Transformation is the term used to describe when a body is repositioned or makes some movement.
Some of the movements involved in transformation are:
Rotation Translation and so onHow to transform the pre- image to the imageThe movement can start in several ways however we stick to this as described
The first movement is rotation by 180 degrees about the topmost edge at the left side.The next step is translation 14 units to the right. This gets the preimage exactly on top of the imageFinally, translation 4 units downLearn more about translation at: https://brainly.com/question/29042273
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Pls help & also give an easy explanation thank youuuuu
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Given
A digital picture frame with a border of 3 cm. The actual length of the frame is x
Answer
a) The actual side of the picture is x-3
Area of picture
[tex]=(side)^2=(x-3)^2[/tex]b) Area of frame
[tex]x^2[/tex]c) Area of border = Area of frame - area of picture
[tex]x^2-(x-3)^2[/tex]Find the equation of the line described. Write your answer in standard form. Vertical and containing (10,14)
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We have here a special case where the line is vertical. In this case, the line has an "infinite" slope (or it is not defined). Therefore, since the line is vertical and contains the point (10, 14), the line is given by the equation:
[tex]x=10[/tex]The standard form of the line is given by the general equation:
[tex]Ax+By=C[/tex]Then, we can rewrite the equation as follows:
[tex]x+0y=10[/tex]We can see that this line contains the point (10,14):
We can see that the vertical line, x + 0y = 10 passes through the point (10, 14).
In summary, the line is given by x + 0y = 10 (A = 1, B = 0, C = 10).
The sum of 4 times a number and 5 is the same as the difference of the number and 7
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Answer:
4nx5=n-7
-7/19
Step-by-step explanation:
Let's Write this equation step by step.
The sum of 4 times a number and 5
That means: 4nx5
The difference of the number and 7
That means: n-7
Now, combine:
4nx5=n-7
-7/19
End Behavior Graphically
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We will investigate how to determine the end behaviours of polynomial functions.
The function given to us is:
[tex]f(x)=123x^3+9x^4-786x-3x^{5^{}}-189x^2\text{ + 1260}[/tex]Whenever we try to determine the end-behaviour of any function. We are usually looking for value of f ( x ) for the following two cases:
[tex]x\to\infty\text{ and x}\to-\infty[/tex]The most important thing to note when dealing with end-behaviour of polynomial functions is that the behaviour is pre-dominantly governed by the highest order term of a polynomial. The rest of the terms are considered small or negligible when considering end-behaviours of polynomials.
The highest order terms in the given function can be written as:
[tex]f(x)=-3x^5[/tex]Then the next step is to consider each case for the value of ( x ) and evaluate the value of f ( x ) respectively.
[tex]\begin{gathered} x\to\infty \\ f\text{ ( }\infty\text{ ) = -3}\cdot(\infty)^5 \\ f\text{ ( }\infty\text{ ) = -3}\cdot\infty \\ f\text{ ( }\infty\text{ ) = -}\infty \end{gathered}[/tex]Similarly repeat the process for the second case:
[tex]\begin{gathered} x\to-\infty \\ f\text{ ( -}\infty\text{ ) = -3}\cdot(-\infty)^5 \\ f\text{ ( -}\infty\text{ ) = 3}\cdot\infty \\ f\text{ ( -}\infty\text{ ) = }\infty \end{gathered}[/tex]Combining the result of two cases we get the following solution:
[tex]As\text{ x}\to\text{ }\infty\text{ , y}\to\text{ -}\infty\text{ and as x}\to-\infty\text{ , y}\to\text{ }\infty[/tex]Correct option is:
[tex]\text{Option C}[/tex]Eighth grad Checkpoint: Understand functions 6NP Which of these relations are functions? Select all that apply. X y 20 -12 12 9 17 2 2013 11 14 -7 3 6 -8 15 16 6 -18 16 15 9 20 15 -9 -5 12 -13 4 20 -18 10 17 13 2. 8 15 Submit
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In a function, any x-value is related to at most 1 y-value.
In the first table, x = 20 and x = 17 are related to 2 different y-values. Then, it is not a function.
In the second table, x = 9 is related to 2 different y-values. Then, it is not a function.
The third and fourth tables are functions
concetta had a 2kg bag of flour. she used 180g of flour to make biscotti. how many kilograms of flour are left in the bag?
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Answer
The amount of flour left in the bag = 1.82 kg
Explanation
Concetta had 2 kg of flour.
She uses 180 g of the flour to make biscotti.
We are then asked to calculate how much flour is left in kilograms.
Amount of flour left = (Initial Amount of flour) - (Amount of flour used)
Initial amount of flour = 2 kg
Amount of flour used = 180 g = 0.18 kg
Amount of flour left = (Initial Amount of flour) - (Amount of flour used)
Amount of flour left = 2 - 0.18 = 1.82 kg
Hope this Helps!!!
Consider the line y= 3/5x-3Find the equation of the line that is parallel to this line and passes through the point (3, 4).Find the equation of the line that is perpendicular to this line and passes through the point (3, 4).
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a) y = 3/5x + 11/5
b) y = -5/3x + 9
Explanation:[tex]\begin{gathered} a)\text{ }y\text{ = }\frac{3}{5}x\text{ - 3} \\ \text{compare with equation of line:} \\ y\text{ = mx + b} \\ m\text{ =slope, b = y-intercept} \\ m\text{ =slope = 3/5} \\ b\text{ = -3} \end{gathered}[/tex]For a line to be parallel to another line. the slope of the 1st line will be equalt to the slope of the 2nd line:
slope of 1st line = 3/5
So, the slope of the 2nd line = 3/5
Given point: (3, 4) = (x, y)
To get the y-intercept of the second line, we would insert the slope and the point into the equation of line
[tex]\begin{gathered} y\text{ = mx + b} \\ 4\text{ = }\frac{3}{5}(3)\text{ + b} \\ 4\text{ = 9/5 + b} \\ 4\text{ - }\frac{\text{9}}{5}\text{ = b} \\ \frac{20-9}{5}\text{ = b} \\ b\text{ = 11/5} \end{gathered}[/tex]The equation of line parallel to y = 3/5x - 3:
[tex]\begin{gathered} y\text{ = mx + b} \\ y\text{ = }\frac{3}{5}x\text{ + }\frac{11}{5} \end{gathered}[/tex][tex]b)\text{ line perpendicular to y = 3/5x - 3}[/tex]For a line to be perpendicular to another line, the slope of one will be the negative reciprocal of the second line
Slope of the 1st line = 3/5
reciprocal of 3/5 = 5/3
negative reciprocal = -5/3
slope of the 2nd line (perpendicular) = -5/3
We need to get the y-intercept of the perpendicular line:
[tex]\begin{gathered} \text{given point: (3,4) = (x, y)} \\ y\text{ = mx + b} \\ m\text{ of the perpendicular = -5/3} \\ 4\text{ = }\frac{-5}{3}(3)\text{ + b} \\ 4\text{ = -5 + b} \\ 4\text{ + 5 = b} \\ b\text{ = 9} \end{gathered}[/tex]The equation of line perpendicular to y = 3/5x - 3:
[tex]\begin{gathered} y\text{ = mx + b} \\ y\text{ = }\frac{-5}{3}x\text{ + 9} \end{gathered}[/tex]What is a quadrilateral that has reflection symmetry, but not rotation symmetry?
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The quadrilaterals, parallelogram,square, rectangle has rotational symmetry but no reflectional symmetry
A trapezoid has neither a rotational symmetry nor a reflectional symmetry
But for an isosceles with only one pair of parallel sides has a reflectional symmetry but no rotational symmetry
Thus, the correct answer is
an isosceles with only one pair of parallel sides
The endpoints are a side of a rectangle ABCD in the coordinate plane at A(3,4), B(6,1) Find the equation of the line the given segment The line segment is line Segment AB
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The endpoints are a side of a rectangle ABCD in the coordinate plane at A(3,4), B(6,1) Find the equation of the line the given segment
The line segment is line Segment AB
step 1
Find the slope of segment AB
m=(1-4)/(6-3)
m=-3/3
m=-1
step 2
Find the equation of the line in slope intercept form
y=mx+b
we have
m=-1
point (3,4)
substitute
4=(-1)*(3)+b
4=-3+b
b=4+3
b=7
therefore
the equation of segment AB is
y=-x+7i inserted a picture of the question can you please list the answers as well
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Solution
We want to find the equation of the line given in the graph
We can see the four points on the graph where the line pass through
The points are
[tex](4,4),(2,3),(0,2),(-4,0)[/tex]We first obtain the slope (m)
The formula for finding the slope is given as
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Using the points (0,2) and (-4,0) (indeed we can pick any two points, we will still obtain the same answer)
Here
[tex]\begin{gathered} x_1=0 \\ y_1=2 \\ x_2=-4 \\ y_2=0 \end{gathered}[/tex][tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{0-2}{-4-0} \\ m=\frac{2}{4} \\ m=\frac{1}{2} \\ m=0.5 \end{gathered}[/tex]We can use any of the points above to find the equation
Equation of a line is given by
[tex]y-y_1=m(x-x_1)[/tex]Using (4,4)
[tex]y-4=0.5(x-4)[/tex]Option D is correct
Using (2,3)
61 less than twice vidy's score
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We are given the following word problem
"61 less than twice Vidy's score"
Let us translate the word problem into an algebraic expression
Let v represents Vidy's score
twice Vidy's score means 2 times v
61 less than means to subtract 61 from 2 times v
So, the algebraic expression becomes
[tex]2v-61[/tex]Conner plans to plow a field in one day. Before lunch he plows 15 acres, which is 30% of the field. Howmany acres will he have to plow after lunch in order to finish the field?
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Solution:
Let the total field to plow be 100 %
According to the question,
The field plowed before lunch is shown below:
15 acres field = 30%
30 % field = 15 acres
1 % field = 15/30 acres
The field plow after lunch is 70%.
[tex]\begin{gathered} 70\text{ \% of field = }\frac{15}{30}\times70 \\ =35\text{ acres} \end{gathered}[/tex]Final Answer:
Therefore, the field to plow after lunch in order to finish the field is 35 acres.
59.25 ÷ 0.75 = 1.06 × 7.3 =on chart. will send image
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For the division, notice that we can multiply both numbers by 100, to get the following:
[tex]\begin{gathered} 59.25\cdot100=5925 \\ 0.75\cdot100=75 \end{gathered}[/tex]then, we can make the long division:
therefore, the result of 59.25 ÷ 0.75 is 79
For the multiplication, we can write the following:
notice that since both factors have 2 digits and 1 digit each after the decimal point, the final result will have 3 digits after the decimal point,
Therefore, the result of 1.06 × 7.3 is 7.738
What are the coefficient(s) in the following expression:
x² + 2x-5xy-y+3y¹
2,4
A
B
C
D
1, 2, 5, 1,3
2,-5, 3
1, 2, 5, 1, 3
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Step-by-step explanation:
based on the expression you wrote here, the correct answer is
1, 2, -5, -1, 3
since none of your answer options show this, you must have made a mistake either with the expression itself or with the answer options.
please choose in your original the one matching my answer above.
If these two figures are similar, what is the measure of the missing angle?
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If the two figures are similar, then the missing angle equals 70°.
State whether the given set of lines are parallel, perpendicular or neither.3x-2y=56y-9x=6The lines are Answer
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Two lines are parallel if:
[tex]m1=m2[/tex]Two lines are perpendicular if:
[tex]m1\cdot m2=-1[/tex]---------------------
Let's rewrite the given equations in the slope-intercept form:
[tex]\begin{gathered} 3x-2y=5 \\ y=\frac{3}{2}x-\frac{5}{2} \\ -------- \\ 6y-9x=6 \\ y=\frac{3}{2}x+1 \end{gathered}[/tex]Since:
[tex]\begin{gathered} m1=m2 \\ \frac{3}{2}=\frac{3}{2} \\ \end{gathered}[/tex]We can conclude that the lines are parallel.
The function P(x) is mapped to I(x) by a dilation in the following graph. Line p of x passes through (negative 2, 4) & (2, negative 2). Line I of X passes through (negative 4, 4) & (4, negative 2).© 2018 StrongMind. Created using GeoGebra. Which answer gives the correct transformation of P(x) to get to I(x)?
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When we're dilating a line, we can either multiply the function value by a constant
[tex]f(x)\to kf(x)[/tex]or the argument of the function
[tex]f(x)\to f(kx)[/tex]Since the y-intercept of both functions is the same, then the multiplied quantity was the argument of the function.
We want to know the constant associated to the transformation
[tex]I(x)\to I(kx)=P(x)[/tex]We have the following values for both functions
[tex]\begin{gathered} I(-4)=4,\:I(4)=-2 \\ P(-2)=4,\:P(2)=-2 \end{gathered}[/tex]For the same y-value, we have the following correlations
[tex]\begin{gathered} I(-4)=P(-2)=P(\frac{1}{2}\cdot-4) \\ I(4)=P(2)=P(\frac{1}{2}\cdot4) \\ \implies I(x)=P(\frac{1}{2}x) \end{gathered}[/tex]and this is our answer.
[tex]I(x)=P(\frac{1}{2}x)[/tex]May you tell me which equation you would choose to solve for one of the variables and explain please.
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This is a simultaneous system of equations, we would need both equations, to solve for the variables.
we have
2x - 3y = 6 - ---i
x +7y = 2------ii
Let's modify equation ii, x +7y = 2 means x = 2 - 7y
Anywhere we see x in equation i, lets put in 2 - 7y instead
2( 2 - 7y) - 3y = 6
4 - 14y - 3y = 6
4 - 17y = 6
-17y = 2
y = -2/17
Lets put this result in equation ii
[tex]undefined[/tex]How do I solve and what would the answer be?
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To find the inverse of a function:
[tex]f(x)=\frac{2}{x-5}[/tex]We will follow the steps below
Step 1: Replace f(x) witth y
[tex]y=\frac{2}{x-5}[/tex]Step 2: interchange x with y
[tex]x=\frac{2}{y-5}[/tex]Step 3: Make y the subject of the formula
[tex]\begin{gathered} y-5=\frac{2}{x} \\ \\ y=\frac{2}{x}+5 \end{gathered}[/tex]Thus, the inverse of the function is:
[tex]y=\frac{2}{x}+5[/tex]in the lab Dale has two solutions that contain alcohol and is mixing them each other.she uses four times as much solution A as solution B.solution a is 20% of alcohol and solution B is 15% of alcohol. how many milliliters of solution B does he use, if the was resulting mixtures has 570 milliliter of pure alchohol.number of milliliters of solution B__?
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Let:
• A ,be the number of millilitres (mL) of solution A used.
,• B ,be the number of mL of solution B used.
We know that Dale uses four times as much solution A as solution B, meaning
[tex]A=4B[/tex]Now, we know that we will end up with 570 mL of pure alcohol in the final solution. Using the dilution of both A and B (20% means 0.2 and 15% is 0.15) we would have that:
[tex]0.2A+0.15B=570[/tex]We would have the following system of equations:
[tex]\begin{cases}A=4B \\ 0.2A+0.15B=570\end{cases}[/tex]Substituting equation 1 in equation 2 and solving for B :
[tex]\begin{gathered} 0.2A+0.15B=570 \\ \rightarrow0.2(4B)+0.15B=570 \\ \rightarrow0.8B+0.15B=570 \\ \rightarrow0.95B=570\rightarrow B=\frac{570}{0.95} \\ \Rightarrow B=600 \end{gathered}[/tex]Substituting in equation 1 and solving for A:
[tex]\begin{gathered} A=4B \\ \rightarrow A=4(600) \\ \Rightarrow A=2400 \end{gathered}[/tex]This way, we can conclude that 2400 mL of solution A and 600mL of solution B were used.