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2 years ago

Which of the following represents the factored form of f(x) = x3 − 64x?

Mathematics

1 answer:

sp2606 [1]2 years ago

8 0

Answer:

x(x + 8)(x - 8)

Step-by-step explanation:

Given

x³ - 64x ← factor out x from each term

= x(x² - 64) ← x² - 64 is a difference of squares

= x(x + 8)(x - 8) ← in factored form

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Step-by-step explanation: answer is B

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and because the ________________________________. Therefore, ABCD is a parallelogram because both pairs of opposite sides are pa

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The answer is "lengths of opposite sides are equal "

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Answer:

Step-by-step explanation:

As the statement is ‘‘if and only if’’ we need to prove two implications

$f : X \rightarrow Y$ is surjective implies there exists a function $h : Y \rightarrow X$ such that $f\circ h = 1_Y$ .
If there exists a function $h : Y \rightarrow X$ such that $f\circ h = 1_Y$ , then $f : X \rightarrow Y$ is surjective

Let us start by the first implication.

Our hypothesis is that the function $f : X \rightarrow Y$ is surjective. From this we know that for every $y\in Y$ there exist, at least, one $x\in X$ such that $y=f(x)$ .

Now, define the sets $X_y = \{x\in X: y=f(x)\}$ . Notice that the set $X_y$ is the pre-image of the element $y$ . Also, from the fact that $f$ is a function we deduce that $X_{y_1}\cap X_{y_2}=\emptyset$ , and because $f$ the sets $X_y$ are no empty.

From each set $X_y$ choose only one element $x_y$ , and notice that $f(x_y)=y$ .

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Now, let us prove the second implication.

We have that there exists a function $h:Y\rightarrow X$ such that $f\circ h=1_Y$ .

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So, for every element $y\in Y$ we have found that an element $x\in X$ (recall that $x=h(y)$ ) such that $y=f(x)$ , which is equivalent to the fact that $f$ is surjective. Therefore, the prove is complete.

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