(4.04 LC)

Which equation represents a linear function?

Equation 1: y3 = 2x + 1
Equation 2: y = 3x + 1
Equation 3: y = 5x2 − 1
Equation 4: y = 4x4 − 1

A. Equation 1
B.Equation 2
C. Equation 3
D. Equation 4

They can differ only by a constant is true of f(x) and g(x) if both are antiderivatives of f(x) Hence, option D is correct.

When two functions, F(x)  and G(x), are antiderivatives of the same function f(x), it means that their derivatives are equal to f(x).

This relationship can be represented as:

F'(x) = G'(x) = f(x)

However, it's important to note that if  F(x) and G(x) are both antiderivatives of F(x), then their difference, F(x) - G(x), will have a derivative of zero.

Consequently, F(x) and G(x) can differ only by a constant.

So, the correct option is D.

Complete question:

What must be true of f(x) and G(x) if both of them are antiderivatives of f(x)?

A. They are the same function

B. They can differ by a factor of x

C. If is not possible for two functions to be antiderivatives of the same function

D. They can differ only by a constant

There are 243 ways to select five elements in order from a set of three elements with repetition allowed.

When selecting five elements in order from a set with three elements and repetition is allowed, each selection can include any of the three elements, repeated as necessary. Here's the breakdown:

1. For the first position, there are 3 choices.

2. For the second position, there are also 3 choices, as repetition is allowed.

3. Similarly, for the third, fourth, and fifth positions, there are 3 choices each.

To find the total number of ways, we multiply the number of choices for each position:

3 choices for the first position × 3 choices for the second position × 3 choices for the third position × 3 choices for the fourth position × 3 choices for the fifth position = [tex]\(3^5 = 243\)[/tex] ways.

Therefore, there are 243 different ways to select five elements in order from a set with three elements when repetition is allowed.

The correct naswer is 21.

The number of ways to select five elements in order from a set with three elements when repetition is allowed can be represented in LaTeX as:

[tex]\binom{5+3-1}{5} = \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = 21[/tex]

Explanation:

- When repetition is allowed, the problem can be treated as finding the number of ways to arrange 5 objects with 3 distinct types.

- This can be solved using the combination formula, where we choose 5 positions out of 7 (5 elements + 3 distinct types - 1).

- The binomial coefficient [tex]\binom{n}{r}[/tex] represents the number of ways to choose [tex]$r$[/tex] items from a set of [tex]$n$[/tex] items.

- In this case, we are choosing 5 positions (elements) from a set of 7 positions (5 elements + 3 distinct types - 1).

- The binomial coefficient can be expanded using factorials: [tex]\binom{n}{r} = \frac{n!}{r!(n-r)!}[/tex]

- Substituting [tex]n = 7$ and $r = 5[/tex], we get[tex]\binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = 21[/tex]

Therefore, there are 21 different ways to select five elements in order from a set with three elements when repetition is allowed.

Whole numbers are sometimes integers. Correct option is b.

Integers include both positive and negative whole numbers, as well as zero. Whole numbers are a subset of integers, but they do not include negative numbers. So, while all whole numbers are integers, not all integers are whole numbers.

Relationship between Whole Numbers and Integers:

Every whole number is an integer: Since whole numbers include zero and all positive counting numbers, they are also part of the set of integers.

Not every integer is a whole number: Integers also include negative numbers, which are not part of the set of whole numbers.

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Answer:

5.2

Step-by-step explanation:

The hypothesis in the given mathematical conditional statement 'If the sum of two angles is 90°, then the angles are complementary.' is 'the sum of two angles is 90°'.

In a conditional statement in mathematics, the 'if' part of the statement is called the hypothesis and the 'then' part is termed the conclusion. Given the statement 'If the sum of two angles is 90°, then the angles are complementary.', the hypothesis of this statement is 'the sum of two angles is 90°'.

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In a conditional statement, the hypothesis is the condition that needs to be met. In this case, the hypothesis of the statement 'If the sum of two angles is 90°, then the angles are complementary,' is 'the sum of two angles is 90°.'

In the context of the given conditional statement, 'If the sum of two angles is 90°, then the angles are complementary,' the hypothesis refers to the clause immediately after 'if.' This indicates the condition that needs to be fulfilled for the conclusion to be considered valid. Therefore, the hypothesis for this statement is 'the sum of two angles is 90°'.

After the 'if,' the hypothesis is given, and after the 'then,' you find the conclusion. The conclusion in this case is 'the angles are complementary.'

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Answer:

I think Table C - Corralle

Answer:

c

Step-by-step explanation:

An underlined greater than sign in mathematics represents a strict inequality, indicating that one value is significantly greater than another.

In mathematics, an underlined greater than sign usually represents an inequality. When a greater than sign (>) is underlined, it indicates a strict inequality, meaning that the value on the left side is significantly greater than the value on the right side.

For example, if we have the underlined inequality 5 > 3, it means that 5 is larger than 3 and there is a clear distinction between the two values.

It's important to note that this is just one possible interpretation of an underlined greater than sign, as the context in which it is used can vary.

Zeros of the given equation [tex]3x^{2} -7x+ 1[/tex] are [tex]\frac{7+\sqrt{37} }{6} \ or \frac{7-\sqrt{37} }{6}[/tex].

The zeros of a quadratic equation f(x) are all the x-values that make the polynomial equal to zero.

The quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula:

[tex]x = \frac{-b \pm \sqrt{b^{2} -4ac} }{2a}[/tex]

According to the given question.

We have a function.

[tex]f(x) = 3x^{2} -7x+1[/tex]

To find the zeros of the function equate f(x) = 0.

[tex]3x^{2} -7x+1 = 0\\[/tex]

Solve the above equation by quadratic method.

[tex]x = \frac{7\pm\sqrt{(7)^{2} -4(3)(1)} }{2(3)}[/tex]

[tex]\implies x = \frac{7\pm\sqrt{49-12} }{6}[/tex]

[tex]\implies x = \frac{7\pm\sqrt{37} }{6}[/tex]

[tex]\implies x = \frac{7+\sqrt{37} }{6} \ or \frac{7-\sqrt{37} }{6}[/tex]

Hence, zeros of the given equation [tex]3x^{2} -7x+ 1[/tex] are [tex]\frac{7+\sqrt{37} }{6} \ or \frac{7-\sqrt{37} }{6}[/tex].

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Answer:

-20/29

Step-by-step explanation:

The expression that satisfies the given condition is using if-else conditional statements to check the value of the variable 'month' and assign the corresponding month name.

The expression that satisfies the given condition is:

if (month == 1) {     answer = "jan"; } else if (month == 2) {     answer = "feb"; } else if (month == 3) {     answer = "mar"; } // ... continue this pattern for the remaining months

This code uses conditional statements (if-else) to check the value of the variable 'month' and assigns the corresponding month name to the variable 'answer'.

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The question concerns a basic algebraic expression 'eight times the sum of a and b' which is represented as 8(a + b). The expression emphasizes the operation order: sum first, then multiply, which results in a value eight times greater than the original sum.

The question 'eight times the sum of a and b' is a mathematical expression that can be represented as 8(a + b). This expression means you first add the numbers 'a' and 'b' and then multiply their sum by eight. The result you get after the multiplication will be eight times greater than the original sum of 'a' and 'b'. For instance, if 'a' is 2 and 'b' is 3, their sum is 5, and when this is multiplied by eight, it becomes 40, which is the desired expression's value.

To understand this concept further, we can refer to the exponentiation rule mentioned, which states that (xa)b = xa.b. Although this is a different type of operation—exponentiation—it demonstrates a similar principle of first performing the operation inside the parentheses and then applying the outside operation.

Finally, when performing algebraic operations, it's essential to remember that whatever you do to one side of the equation, you should do to the other side to maintain balance. This is a fundamental principle in algebra that helps to solve equations, such as the example provided showing how to isolate 'a' by subtracting 'x' from both sides of the equation a b.

2

2x2 = 4

2+2 =4

0

0x0=0

0+0=0