All categories

3 years ago

Use the functions f(x)=2x and g(x)=x^2+1 to find the value of each expression

Mathematics

1 answer:

Alinara [238K]3 years ago

6 0

Answer:

1. 20

2. 23

3. 6

Step-by-step explanation:

We have that:

f(x) = 2x

g(x) = x² + 1

f(g(x)) is the composite function of f and g. So

f(g(x)) = f(x²-1) = 2(x²+1) = 2x² + 2

1. f(g(3))

f(g(x)) = 2x² - 2 = 2(3)² + 2 = 18 + 2 = 20

2. f(3)+g(4)

f(3) = 2(3) = 6

g(4) = 4² + 1 = 17

f(3) + g(4) = 6 + 17 = 23

3. f(5) - 2g(1)

f(5) = 2(5) = 10

g(1) = (1)² + 1 = 2

f(5) - 2g(1) = 10 - 2*2 = 10 - 4 = 6

You might be interested in

3. An aspirin tablet weights 0.5 grams. How many ounces does it weigh?

maksim [4K]

3. An aspirin weighs 0.018 ounces.
1 gram = 0.035 ounces

4. An aspirin is 500 milligrams.
1,000 milligrams = 1 gram

7 0

3 years ago

What two numbers add to equal 11 and multiply to equal 10

Dominik [7]

Hi There! The answer is 10 and 1. Because you add 10 and 1 to get 11 and 10 times 1 equals 10. Hope this helps.

8 0

3 years ago

consider the quadratic form q(x,y,z)=11x^2-16xy-y^2+8xz-4yz-4z^2. Find an orthogonal change of variable that eliminates the cros

Bezzdna [24]

Answer:

$q(x,y,z)=16x^{2}-5y^{2}-5z^{2}$

Step-by-step explanation:

The given quadratic form is of the form

$q(x,y,z)=ax^2+by^2+dxy+exz+fyz$ .

Where $a=11,b=-1,c=-4,d=-16,e=8,f=-4$ .Every quadratic form of this kind can be written as

$q(x,y,z)={\bf x}^{T}A{\bf x}=ax^2+by^2+cz^2+dxy+exz+fyz=\left(\begin{array}{ccc}x&y&z\end{array}\right) \left(\begin{array}{ccc}a&\frac{1}{2} d&\frac{1}{2} e\\\frac{1}{2} d&b&\frac{1}{2} f\\\frac{1}{2} e&\frac{1}{2} f&c\end{array}\right) \left(\begin{array}{c}x&y&z\end{array}\right)$

Observe that $A$ is a symmetric matrix. So $A$ is orthogonally diagonalizable, that is to say, $D=Q^{T}AQ$ where $Q$ is an orthogonal matrix and $D$ is a diagonal matrix.

In our case we have:

$A=\left(\begin{array}{ccc}11&(\frac{1}{2})(-16) &(\frac{1}{2}) (8)\\(\frac{1}{2}) (-16)&(-1)&(\frac{1}{2}) (-4)\\(\frac{1}{2}) (8)&(\frac{1}{2}) (-4)&(-4)\end{array}\right)=\left(\begin{array}{ccc}11&-8 &4\\-8&-1&-2\\4&-2&-4\end{array}\right)$

The eigenvalues of $A$ are $\lambda_{1}=16,\lambda_{2}=-5,\lambda_{3}=-5$ .

Every symmetric matriz is orthogonally diagonalizable. Applying the process of diagonalization by an orthogonal matrix we have that:

$Q=\left(\begin{array}{ccc}\frac{4}{\sqrt{21}}&-\frac{1}{\sqrt{17}}&\frac{8}{\sqrt{357}}\\\frac{-2}{\sqrt{21}}&0&\sqrt{\frac{17}{21}}\\\frac{1}{\sqrt{21}}&\frac{4}{\sqrt{17}}&\frac{2}{\sqrt{357}}\end{array}\right)$

$D=\left(\begin{array}{ccc}16&0&0\\0&-5&0\\0&0&-5\end{array}\right)$

Now, we have to do the change of variables ${\bf x}=Q{\bf y}$ to obtain

$q({\bf x})={\bf x}^{T}A{\bf x}=(Q{\bf y})^{T}AQ{\bf y}={\bf y}^{T}Q^{T}AQ{\bf y}={\bf y}^{T}D{\bf y}=\lambda_{1}y_{1}^{2}+\lambda_{2}y_{2}^{2}+\lambda_{3}y_{3}^{2}=16y_{1}^{2}-5y_{2}^{2}-5y_{3}^2$

Which can be written as:

$q(x,y,z)=16x^{2}-5y^{2}-5z^{2}$

4 0

3 years ago

On a street map the vertices of a block are w(20,),x(90,30),y(90,120), and z(20,120). The coordinates are measured in yards find

max2010maxim [7]

Answer:

$\text{Perimeter of wxyz}=320$

$\text{Area of the block}=6300\text{ yards}^2$

Step-by-step explanation:

We have been give that one a street map the vertices of a block are w(20,30), x(90,30), y(90,120), and z(20,120). The coordinates are measured in yards. We are asked to find the perimeter and area of the block.

First of all, we will plot our given points on coordinate plane as shown in the attachment.

We can see that block wxyz is in form of a rectangle. We know that perimeter of rectangle is two times the sum of length and width.

The length of the rectangle will be length of segment wx that is the difference between x-coordinates of x and w.

$\text{Length of segment wx}=90-20$