On Saturday, Lukas drove 4x – 5 miles. On Sunday, he drove 3x – 10 miles. What is the total amount of miles lukas drove

If(x) = x + 2 and h(x) = x-1, what is f • h](-3)?

deff fn [24]

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.

4 0

2 years ago

FIND AC! PLS HELP!!

Anna007 [38]

Answer:

AC = 7.3 in

Step-by-step explanation:

tan(51°) = 9/AC

1.24 = 9/AC

AC = 9/1.24

AC = 7.26

5 0

2 years ago

How many dots would be in the 7th picture ?

neonofarm [45]

Answer: 28 dots

Step-by-step explanation: In this pattern,

we can see the the first figure is just 1 dot.

The second figure has a new row on the bottom with 2 dots.

The third figure has a new row on the bottom with 3 dots

and the fourth figure has a new row on the bottom with 4 dots.

So continuing with this pattern,

the next figure will have a new row with 5 dots,

the next will have a new row with 6 dots,

and the next will have a new row with 7 dots.

So in the 7th picture, we will have 7 + 6 + 5 + 4 + 3 + 2 + 1 dots.

This simplifies to 28 dots.

Take a look below.

6 0

3 years ago

Find the arc length of the semicircle. Either enter an exact answer in terms of π or use 3.14 and enter your answer as a decimal

cricket20 [7]

Answer:

The arc length of the semicircle is $5 \pi\ units$ or $15.7\ units$

Step-by-step explanation:

The complete question in the attached figure

we know that

The circumference of a semicircle is given by the formula

$C=\frac{1}{2}\pi D$

where

D is the diameter of the semicircle

we have

$D=10\ units$

substitute

$C=\frac{1}{2}\pi (10)$

$C=5 \pi\ units$ ----> exact value

Assume

$\pi =3.14$

$C=5(3.14)=15.7\ units$ ----> approximate value

therefore

The arc length of the semicircle is $5 \pi\ units$ or $15.7\ units$

6 0

2 years ago

Someone help me with this ples with steps

SSSSS [86.1K]

Answer:

$x=\frac{21}{4}$

$y=\frac{21}{2}$

Step-by-step explanation:

equation (1): $\frac{1}{2} (2x+y)=\frac{21}{2}$

equation (2): y=2x

solve the system by substitution

step1: sub (2x) for y in equation 1:

$\frac{1}{2} (2x+y)=\frac{21}{2}$

$\frac{1}{2} (2x+2x)=\frac{21}{2}$

step2: simplify the expression:

$\frac{1}{2} (4x)=\frac{21}{2}$

$2x=\frac{21}{2}$

(divide by 2 for both sides)

$x=\frac{21}{4}$

To find the value for y sub 21/4 for x in equation 2:

y=2x

y=2×(21/4)

$y=\frac{21}{2}$

4 0

3 years ago