Fiona has proved that a function, f(x), is an arithmetic sequence. How did she prove that?

She showed that an explicit formula could be created.

She showed that a recursive formula could be created.

She showed that f(n) ÷ f(n - 1) was a constant ratio.

She showed that f(n) - f(n - 1) was a constant difference.

Answer:

Quadratic formula for given equation is [tex]x=\frac{-(-1)\pm\sqrt{(-1)-4(7)(-9)}}{2\times7}[/tex]

Step-by-step explanation:

Given Quadratic Equation is  7x² = 9 + x

We need to find correct Quadratic formula for the given quadratic equation.

If the quadratic equation is in standard for,

ax² + bx + c = 0

then quadratic formula is given by,

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

First we rewrite the quadratic equation,

7x² - x - 9 = 0

by comparing with standard form of equation we get,

a = 7  ,   b = -1  and c = -9

oNw putting these value in quadratic formula we get,

[tex]x=\frac{-(-1)\pm\sqrt{(-1)-4(7)(-9)}}{2\times7}[/tex]

Therefore, Quadratic formula for given equation is [tex]x=\frac{-(-1)\pm\sqrt{(-1)-4(7)(-9)}}{2\times7}[/tex]

When we rotate a figure and there is no change in the shape of the figure then it has rotational symmetry.

We know that the order of rotation for a square is 4.

Hence, we have [tex]\frac{360}{4} =90^{\circ}[/tex]

Thus, the angle of rotational symmetry of square are

[tex]90^{\circ}, 180^{\circ}, 270^{\circ}[/tex]

Hence, the minimum angle  of rotational symmetry is  [tex]90^{\circ}[/tex]

Therefore, the minimum angle of rotational symmetry for a square is 90 degrees.

To find the limit of the given expression [(ln(x+5) - ln5) / x ] as x approaches 0, we can apply L'Hospital's Rule.

To find the limit of the given expression, we can use L'Hospital's Rule. Let's first rewrite the expression:

[ln(x+5) - ln(5)] / x

As x approaches 0, when plugging in 0, we get an indeterminate form of 0/0. So, we can apply L'Hospital's Rule:

lim [(ln(x+5) - ln5) / x ] as x approaches 0 = lim [(1/(x+5)) / 1]

= lim (1 / (x+5)) as x approaches 0

Since x approaches 0, the expression approaches 1 / (0+5) = 1/5.

Answer: The simplified form of 48 over 192 is 1 over 4.

Step-by-step explanation:

Since, A fraction is in its simplest form when the numerator and denominator cannot be any smaller, while still being whole numbers.

Also, for find the simplified form of a fraction,

We divide both numerator and denominator by their H.C.F,

Here, the given fraction is,

[tex]\frac{48}{192}[/tex]

Since, 48 = 2 × 2 × 2 × 2 × 3

192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

H.C.F(48,192) =  2 × 2 × 2 × 2 × 3  = 48

[tex]\implies \frac{48/48}{192/48}=\frac{1}{4}[/tex]

Hence, The simplified form of 48/192 is 1/4.

The simplified form of 48/192 is 1/4.

First, let's find the GCD of 48 and 192.

The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The factors of 192 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192.

The common factors are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

So, the greatest common divisor (GCD) of 48 and 192 is 48.

Now, we divide both the numerator and denominator by 48:

48/48 = 1

192/48 = 4

Therefore, the simplified form of 48/192 is 1/4.

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Answer:

Step-by-step explanation:

A relation from a set X to a set Y is called a function, if each element of X is related to one and exactly one element in Y. Though many elements of X can be related to one element in Y (many to one).

{(3, 7),(3, 6),(5, 4),(4, 7)} → is not a Function (as (3, 7), (3, 6) share same x)

{(1, 5),(3, 5),(4, 6),(6, 4)} → is a Function

{(2, 3),(4, 2),(4, 6),(5, 8)} → is not a Function (as (4, 2), (4, 6) share same x)

{(0, 4),(3, 2),(4, 2),(6, 5)} → is a Function

Answer: ^^^ that person was correct

Step-by-step explanation:

Answer: Answer is second and fifth option.

Step-by-step explanation:  

m = –2 and b =

m = –2 and b =  

T he end behavior of f(x) can be described as:

As x → -∞, f(x) → +∞

As x → +∞, f(x) → +∞

To determine the end behavior of f(x) = (x − 3)²(x + 5)(x − 2)³, we need to consider the degree and leading coefficient of the polynomial.

1. Degree of the Polynomial:

The degree is the highest sum of exponents of the variable x in any term. In this case:

(x - 3)² contributes a degree of 2 (x² term)

(x + 5) contributes a degree of 1 (x term)

(x - 2)³ contributes a degree of 3 (x³ term)

Adding these degrees, we get a total degree of 2 + 1 + 3 = 6.

2. Leading Coefficient:

The leading coefficient is the coefficient of the term with the highest degree. Since all factors have a leading coefficient of 1, the leading coefficient of the entire polynomial will also be 1 (positive).

3. End Behavior Based on Degree and Leading Coefficient:

Even Degree & Positive Leading Coefficient: When the degree is even and the leading coefficient is positive, both ends of the polynomial will approach positive infinity. In other words, as x approaches both positive and negative infinity, f(x) will also approach positive infinity.

Therefore, the end behavior of f(x) can be described as:

As x → -∞, f(x) → +∞

As x → +∞, f(x) → +∞

Complete question:

What is the end behavior of [tex]f(x) = (x - 3)^{2} (x + 5)(x - 2)^{3}[/tex]?

The value of the positive integer is,

⇒ 11

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

Here, 12 is added to the square of a positive integer, the result is 133.

Let a positive integer = x

Hence, We can formulate;

⇒ 12 + x² = 133

⇒ x² = 133 - 12

⇒ x² = 121

⇒ x = √121

⇒ x = 11

Thus, The value of the positive integer is,

⇒ 11

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Answer:

The correct option is 3.

Step-by-step explanation:

Total digits 0 to 9 are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Total number of digits is 10. It means total possible ways to select the first digit of the password is 10.

Since the reparation is allowed, therefore the possible ways to select the second and third digit of the password is 10.

Total possible ways to form a three-digit passwords is

[tex]10\times 10\times 10=1000[/tex]

Therefore the correct option is 3.

There are 1,000 possible three-digit passwords that can be formed using the digits 0 through 9 when repetition of digits is allowed, calculated as 10 possibilities for each of the three positions (10 x 10 x 10).

To determine how many possible three-digit passwords can be created using the digits 0 through 9 with repetition allowed, we consider each of the three positions in the password separately. For each position, we can use any of the 10 digits (0-9), leading to 10 possibilities for each digit.

Therefore, the total number of combinations for a three-digit password is:

By the rule of product, we multiply the possibilities for each digit:

10  imes 10  imes 10 = 1,000

Thus, there are 1,000 possible three-digit passwords when digits can be repeated.

Final answer:

The square root of 100 is rational, as is the square root of 0.25.

Explanation:

The square root of 100 is a rational number. A rational number is a number that can be expressed as a quotient or fraction of two integers, where the denominator is not zero. In this case, the square root of 100 is 10, which can be written as 10/1, making it a rational number.

The square root of 0.25 is also a rational number. The square root of 0.25 is 0.5, which can be expressed as 1/2, making it a rational number.

Answer:

It enables you to explore and know more

Step-by-step explanation:

Purpose Of Questionnaire

The main purpose of a questionnaire is to extract data from the respondents. It's a relatively inexpensive, quick, and efficient way of collecting large amount data even when the researcher isn't present to collect those responses first hand

Step-by-step explanation:

It is given that, the total distance an object travels varies directly with the length of time it travels.

[tex]d\propto t[/tex]

If an object travels 32 meters in 18 seconds so,

[tex]32= 18k[/tex].....(1) (k is a constant)

If distance travelled is 80 meters, then the time taken is t say.

[tex]80= kt[/tex].....(2)

On solving equation (1) and (2), we get the value of t as,

t = 45 seconds

Hence, this is the required solution.

we have

[tex]f(x)=(x-4)(x+1)[/tex]

using a graph tool  

see the attached figure

we know that

The function is increasing for all real values of x where [tex]x > 1.5[/tex]

and

The function is decreasing for all real values of x where [tex]x < 1.5[/tex]

therefore

Statements

case a) The function is increasing for all real values of x where x < 0

The statement is false

case b) The function is increasing for all real values of x where x < –1 and where x > 4

The statement is false

case c) The function is decreasing for all real values of x where –1 < x < 4

The statement is false

case d) The function is decreasing for all real values of x where x < 1.5

The statement is true

Answer:

The correct option is:

The function is decreasing for all real values of x where x < 1.5.

Step-by-step explanation:

We are given a function f(x) which is a product of two linear polynomial as:

[tex]f(x)=(x-4)(x+1)[/tex]

Now, we know that on multiplying the two linear polynomial we will obtain a quadratic polynomial.

So, the function f(x) will be represented as:

[tex]f(x)=x(x+1)-4(x+1)\\\\f(x)=x^2+x-4x-4\\\\f(x)=x^2-3x-4[/tex]

So, we will plot the graph of the function and check which statements about the function hold true.

1)

The function is increasing for all real values of x where x < 0.

This statement is false.

Since we get from the graph that function f(x) is decreasing for x<0.

2)

The function is increasing for all real values of x where x < –1 and where x > 4.

This option is incorrect as the function is decreasing for x<-1

whereas it is increasing for x>4.

3)

The function is decreasing for all real values of x where –1 < x < 4.

This option is incorrect.

Since, the function is both decreasing as well as increasing in the interval (-1,4).

4)

The function is decreasing for all real values of x where x < 1.5.

This option is correct.

Since it could be observed from the graph that the function is decreasing for x<1.5.