Answer/Step-by-step explanation:
Composition functions are functions that combine to make a new function. We use the notation ◦ to denote a composition.
f ◦ g is the composition function that has f composed with g. Be aware though, f ◦ g is not
the same as g ◦ f. (This means that composition is not commutative).
f ◦ g ◦ h is the composition that composes f with g with h.
Since when we combine functions in composition to make a new function, sometimes we
define a function to be the composition of two smaller function. For instance,
h = f ◦ g (1)
h is the function that is made from f composed with g.
For regular functions such as, say:
f(x) = 3x
2 + 2x + 1 (2)
What do we end up doing with this function? All we do is plug in various values of x into
the function because that’s what the function accepts as inputs. So we would have different
outputs for each input:
f(−2) = 3(−2)2 + 2(−2) + 1 = 12 − 4 + 1 = 9 (3)
f(0) = 3(0)2 + 2(0) + 1 = 1 (4)
f(2) = 3(2)2 + 2(2) + 1 = 12 + 4 + 1 = 17 (5)
When composing functions we do the same thing but instead of plugging in numbers we are
plugging in whole functions. For example let’s look at the following problems below:
Examples
• Find (f ◦ g)(x) for f and g below.
f(x) = 3x + 4 (6)
g(x) = x
2 +
1
x
(7)
When composing functions we always read from right to left. So, first, we will plug x
into g (which is already done) and then g into f. What this means, is that wherever we
see an x in f we will plug in g. That is, g acts as our new variable and we have f(g(x)).
g(x) = x
2 +
1
x
(8)
f(x) = 3x + 4 (9)
f( ) = 3( ) + 4 (10)
f(g(x)) = 3(g(x)) + 4 (11)
f(x
2 +
1
x
) = 3(x
2 +
1
x
) + 4 (12)
f(x
2 +
1
x
) = 3x
2 +
3
x
+ 4 (13)
Thus, (f ◦ g)(x) = f(g(x)) = 3x
2 +
3
x + 4.
Let’s try one more composition but this time with 3 functions. It’ll be exactly the same but
with one extra step.
• Find (f ◦ g ◦ h)(x) given f, g, and h below.
f(x) = 2x (14)
g(x) = x
2 + 2x (15)
h(x) = 2x (16)
(17)
We wish to find f(g(h(x))). We will first find g(h(x)).
h(x) = 2x (18)
g( ) = ( )2 + 2( ) (19)
g(h(x)) = (h(x))2 + 2(h(x)) (20)
g(2x) = (2x)
2 + 2(2x) (21)
g(2x) = 4x
2 + 4x (22)
Thus g(h(x)) = 4x
2 + 4x. We now wish to find f(g(h(x))).
g(h(x)) = 4x
2 + 4x (23)
f( ) = 2( ) (24)
f(g(h(x))) = 2(g(h(x))) (25)
f(4x
2 + 4x) = 2(4x
2 + 4x) (26)
f(4x
2 + 4x) = 8x
2 + 8x (27)
(28)
Thus (f ◦ g ◦ h)(x) = f(g(h(x))) = 8x
2 + 8x.
A car is 200 km from its destination after 1 h and 80 km from its destination after 3 h.
Answer:
140km after 2 hours
Step-by-step explanation:
Answer:
2=140
Step-by-step explanation:
if 3 sides of a triangle are 9, 40, and 41, which side is the hypotenus?
A.41
B.40
C.9
Answer:
c
Step-by-step explanation:
because hypotenuse is greater than two sides
Answer:
the answer is 41
Step-by-step explanation:
because I love math
Karen has a gold bracelet that weighs 24 grams. She wants to sell the bracelet, but she needs a minimum of one ounce of gold to sell it. Can Karen sell her bracelet? Why or why not?
Answer:
It should be 28.35
Step-by-step explanation:
Hope this helped have an amazing day/month/year!
I can't find the volume
Answer:
8π(2.5)^2
Step-by-step explanation:
volume = π * r^2 * h
v = π * (2.5)^2 * 8
ANSWER ASAP PLEASE
Timothy was trying to find the measure of
above. His answer was 7º. He is confused as he cannot
understand why m
Justify your answer.
Answer:
jim is correct, x does equal 7°. justify your answer by putting seven back into the equation. it gives you back that 180 equals 180, therefore it is correct
Step-by-step explanation:
so, all angles of a triangle have to add up to 180;
(15x) + (8x - 11) + (3x + 9) = 180
15x + 8x - 11 + 3x + 9 = 180
add your common numbers
26x - 2 = 180
add 2
26x = 182
divide 26
x = 7
put it back into the equation
15(7) + 8(7) - 11 + 3(7) + 9 = 180
180 = 180
can i get help please i dont understand this
Answer:
L = 60°M = 50°N = 30°P = 40°Step-by-step explanation:
inside angle = 180 = 2x + 2x + 20 + 3x - 10 + 2x - 10
180 - 20 + 10 + 10 = 2x + 2x + 3x + 2x180 = 9xx = 180 / 9x = 20substitute the value of x = 20 into angle L, M, N, P
L = 2x + 20
L = 2(20) + 20
L = 60°M = 3x - 10
M = 3(20) - 10
M = 50°N = 2x - 10
N = 2(20) - 10
N = 30°P = 2x
P = 2(20)
P = 40°check:∠L + ∠M + ∠N + P = 180
60° + 50° + 30° + 40° = 180
Combine like terms:3n-10+1-4n
Answer:
-n - 9
Step-by-step explanation:
3n and -4n added will just equal to -1n or -n. Add 1 from -10 to get -9. Your final answer should be -n - 9.
Answer:
-n-9
Step-by-step explanation:
3n-10+1-4n
(3n-4n)+(-10+1)
-n-9
Which expressions are equal to 105?
(10'4)1
10'3.10'2
1/10'5
(10'3)2
10.10.10.10.10
Answer:
103 and 102
Step-by-step explanation:
u add them together
For the function f(x) = x2 + 3, find the slope of the tangent line at x = 11.
Given:
The function is
[tex]f(x)=x^2+3[/tex]
To find:
The slope of the tangent line at x = 11.
Solution:
The slope of the tangent is the value of f'(x) at x=11.
We have,
[tex]f(x)=x^2+3[/tex]
Differentiate with respect to x.
[tex]f'(x)=2x+0[/tex] [tex]\left[\because \dfrac{d}{dx}x^n=nx^{n-1},\dfrac{d}{dx}C=0,\text{ where C is constant}\right][/tex]
[tex]f'(x)=2x[/tex]
Substitute x=11 in the above equation.
[tex]f'(11)=2(11)[/tex]
[tex]f'(11)=22[/tex]
Therefore, the slope of the tangent at x=11 is 22.
help need plzzzzzzzzz
- a midline of a trapezoid is always half of the sum of the legs.
so,
2TW = UV + SX
now plug in and solve,
2(14p + 22) = 17p + 4p + 65
28p + 44 = 21p + 65
7p = 21
p = 3
request: please mark brainiest, and click the thanks button if correct <3
What is the vertical line test?
Basically, in a function, you draw a vertical line. If that vertical line crosses the function more than once, then it is an invalid function.
Let me know if you need an example and I'll edit a picture in for you.
Answer:
A vertical line test is used to find out whether if a curve is a graph of a function or not. If you are able to draw a vertical line, and it intersects more than one point on the curve, then it is not a function.
Step-by-step explanation:
katie pays $725 a month for rent, $40 for cable and $3.60 an hour for local telephone service. if his monthly expenditures last month were $826.20 how much time did he spend on the phone.
PLEASE HELP,, MARKING BRAINLIEST!!!
Given coordinates N(-1,-11) and P(-1,-3), find NP?
Given:
Two points are N(-1,-11) and P(-1,-3).
To find:
The value of NP.
Solution:
Distance formula:
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Two points are N(-1,-11) and P(-1,-3). Using distance formula, we get
[tex]D=\sqrt{(-1-(-1))^2+(-3-(-11))^2}[/tex]
[tex]D=\sqrt{(-1+1)^2+(-3+11)^2}[/tex]
[tex]D=\sqrt{(0)^2+(8)^2}[/tex]
On further simplification, we get
[tex]D=\sqrt{0+64}[/tex]
[tex]D=\sqrt{64}[/tex]
[tex]D=8[/tex]
Therefore, the value of NP is 8 units.
In a survey of 2,500 people who owned a certain type of car, 1,000 said they would buy that type of car again. What percent of the people surveyed were satisfied with the car?
Answer:
40%
Step-by-step explanation:
Given that :
Number of people surveyed = 2500
Number of satisfied buyers = 1000
Thus, the percentage of satisfied buyers :
(Satisfied buyers / total surveyed) * 100%
(1000 / 2500) * 100%
0.4 * 100 = 40%
40% of people surveyed were satisfied
Please answer this!!
Answer:
Step-by-step explanation:
Exercise 1
16/24
Exercise 2
3:6
Exercise 3
5:3
Exercise 4
3:20
Exercise 5
10:3
which of the following is a point on the graph of y=(1/2)x
Answer:
left, hope this helps
Step-by-step explanation:
(20,-3),(19,-20) what is the slope of this
slope = 17
- use the formula for slope, and simply plug it in.
Look at this diagram.
If KM and NP are parallel lines and m<MLJ = 42°, what is m<MLO?
____°
9514 1404 393
Answer:
138°
Step-by-step explanation:
The angles of the linear pair are supplementary.
∠MLO = 180° -∠MLJ = 180° -42°
∠MLO = 138°
Answer:
angle MLO =180° - angle MLJ =(180°-42°)= 138°
138° is the right answer.1. The measure of side c is 150 feet and the
measure of side a is 60 feet. What is the
length of side b?
Answer:
Hi...
a^2+b^2=c^2
b^2=c^2-a^2
b^2=22500-3600=18900
So...b=137/4
A box contains 24 cookies. Eighteen cookies are chocolate chip
cookies. Another box contains 36 cookies. How many cookies
in this box would have to be chocolate chip so that the ratios of
chocolate chip cookies to total cookies in the boxes are the same?
Hi can someone pls help me with this
Answer:
these all are in browser u can get them from searching in browser
i need help with this problem plz
Answer:
-1
Step-by-step explanation:
your welcome and ignore my bad writing
Please help me if you are good at math!
PLEASE HELP
...........
what is the rate of change of the function represented by this graph?
Answer:
It's 0
Step-by-step explanation:
The price of a computer was $ 375. In a sale, the price was reduced by 15%. Calculate the reduction in the price of the computer.
Answer:
318.75
Step-by-step explanation:
15% of 375 is 56.25. then you just subtract 56.25 from 375 and get 318.75
Answer:
$ 56.25
Step-by-step explanation:
Reduction = 15% of 375
[tex]= \frac{15}{100}*375\\\\= 0.15*375\\= $56.25[/tex]
PLZ I NEED HELP. WILL GIVE BRAINLIEST
What will the sign be on the solution to this problem: (-3)(-4)(5)(6)(-7)
-
+
Answer:
it will be -.
Step-by-step explanation:
The slope of a line is -5 and its y-intercept is 8. Discuss how you will write the equation in
the slope- intercept form.
Answer:
the formula = y = mx+c,
if slope is -5 and y intercept is 8,
y = -5x+8
or
y= 8-5x
Please help I don’t get it
Answer:
1.03
Step-by-step explanation:
10.52-1.25=9.27
9.27/3=1.03
how do I turn 3 1/5 into a decimal?
Answer:
3.2
Step-by-step explanation:
Step 1: Multiply the whole number by the denominator: 3 × 5 = 15.
Step 2: Add the product you got in Step 1 to the numerator: 15 + 1 = 16.
Step 3: Divide the sum from Step 2 by the denominator: 16 ÷ 5 = 3.2. That's it folks!
Answer:
3.2
Step-by-step explanation:
Step 1: divide numerator (1) by the denominator (5): 1 ÷ 5 = 0.2. Step 2: add this value to the the integer part: 3 + 0.2 = 3.2. So, 3 1 / 5 = 3.2.