If f(x) is a continuous function defined for all real numbers, f(-2) = 10, f(-7) = -5, and f(x) = 0 for one and only one value of x, then which of the following could be that x value? a) -3 b) -8 c) 0 d) 2

Answers

Answer 1

Answer:

Step-by-step explanation:

The given information tells us that f(x) is a continuous function, and it passes through the points (-2, 10) and (-7, -5). Additionally, it has exactly one value of x for which f(x) = 0. We need to find which of the given options (-3, -8, 0, 2) could be that x value.

Let's analyze each option:

a) -3: If f(-3) = 0, it's a possibility, but we need to check the others.

b) -8: If f(-8) = 0, it's a possibility, but we need to check the others.

c) 0: If f(0) = 0, it's a possibility, but we need to check the others.

d) 2: If f(2) = 0, it's a possibility, but we need to check the others.

Since we are told that there is only one value of x for which f(x) = 0, and there are multiple possibilities in this case (a, b, c, and d), the information provided does not allow us to determine a unique answer from the given options. Any of these values could be the x for which f(x) = 0, given that the function is continuous.

Answer 2

Answer:

Final answer:

The question uses the concept of Intermediate Value Theorem in mathematics. The answer is option a) -3 as it falls between the range by the theorem, which states that for any continuous function within a certain range, the function must cross all values within that range.

Explanation:

This question is about the Intermediate Value Theorem, which states that if a function is continuous on a closed interval from a to b, and k is any number between f(a) and f(b), then there is at least one number c in the interval (a, b) such that f(c) = k.

Since the function is continuous for all real numbers and f(-2) = 10 and f(-7) = -5, the function must pass through 0 between -2 and -7 as it moves from 10 to -5. Therefore, the x value for which f(x) = 0 could be between -2 and -7. In the provided options, the only value falling within the interval (-2, -7) is -3.

So, based on the Intermediate Value Theorem, the answer is option a) -3.

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3. Given the following functions f(x) and g(x), solve f[g(7)] and select the correct answer below:f(x) = 4x + 21

g(x) = 2x + 2

16
49
85
784

Answers

The value of the required function f[g(7)] is 85.

What is a function?

A function is a relation between the input and the output of a system.

Given functions are:

$f(x)= 4x + 21\ng(x)=2x+2$

Therefore, the required function:

$f(g(x))\n= 4g(x)+21\n= 4(2x+2)+21\n= 8x+8+21\n= 8x+29$

Now,

$f[g(7)]\n= 8(7)+29\n= 56+29\n= 85$

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Hello,
f(g(x))=f(2x+2)=4(2x+2)+21=8x+29
f(g(7))=8*7+29=56+29=85

Define powers provide an example of power with an

Answers

An exponent, or a power, is mathematical shorthand for repeated multiplications. An example of a power with an exponent of 3 is 2³, which equals 8.

An exponent, or a power, is mathematical shorthand for repeated multiplications. For example, the exponent "2" means to multiply the base for that exponent by itself.

So, for an exponent of 3, the base would be multiplied by itself three times.

An example of a power with an exponent of 3 would be 2³. This means that you need to multiply the base, which is 2, by itself three times: 2 x 2 x 2 = 8.

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The probable question may be:

What is the definition of powers in mathematics, and can you provide an example of a power with an exponent of 3?

These two functions represent the growth of two different bacterial cultures in terms of the number of bacteria after x days.f(x) = 2,000(2)^x (other one on the graph)

Answer the following:
A) Which function has the higher initial amount of bacteria? g(x) or f(x)
B) Which function has the greater amount of bacteria after two days? g(x) or f(x)

Answers

Answer:

A. The function f(x) has the higher initial amount of bacteria.

B. The function g(x) has the higher amount of bacteria after two days.

Step-by-step explanation:

The given function is

$f(x)=2000(2)^x$

The graph of g(x) passing through the points (0,1000) and (1,3000). So the initial value is 1000 and the growth factor is 3.

The function g(x) is

$g(x)=1000(3)^x$

Part A:

Substitute x=0, to find the initial blue of the functions.

$f(0)=2000(2)^0=2000$

$g(0)=1000(3)^0=1000$

Since 2000>1000, therefore the function f(x) has the higher initial amount of bacteria.

Part B:

Substitute x=2, to find the amount of bacteria after two days.

$f(2)=2000(2)^2=8000$

$g(2)=1000(3)^2=9000$

Since 8000<9000, therefore the function g(x) has the higher amount of bacteria after two days.

Find the equation of the line through (2, 3, 0) perpendicular to the vectors a = (1, 2, 3) and b = (3, 4, 5).

Answers

The cross product a×b is a vector perpendicular to both. That cross product is
(1, 2, 3)×(3, 4, 5) = (-2, 4, -2)

This can have common factors removed and be sign-adjusted to (1, -2, 1), so the equation of the line can be written in parametric form as ...
(x, y, z) = (2, 3, 0) + t·(1, -2, 1)

_____
When working with vectors and matrices, a suitable calculator can be very helpful. There are several phone and tablet apps and web sites that can do these calculations for you.

Make x the subject of the formula $r = \sqrt{ (ax - p)/(q + bx) }$

Answers

Answer:

$\boxed{x = \frac{p + q {r}^(2) }{a - b {r}^(2) } }$

Step-by-step explanation:

$Solve \: for \: x: \n = > r = \sqrt{ (ax - p)/(q + bx) } \n \n </p><p>r = \sqrt{ (ax - p)/(q + bx) } \: is \: equivalent \: to \: \sqrt{ (ax - p)/(q + bx) } = r :\n = > \sqrt{ (ax - p)/(q + bx) } = r \n \n Raise \: both \: sides \: to \: the \: power \: of \: two: \n = > (ax - p)/(q + bx) = {r}^(2) \n \n Multiply \: both \: sides \: by \: (q + b x): \n = > ax - p = {r}^(2) (q + bx) \n \n Expand \: out \: terms \: of \: the \: right \: hand \: side: \n = > ax - p = q {r}^(2) + b {r}^(2)x \n \n Subtract \: b {r}^(2)x - p \: from \: both \: sides: \n = > x(a - b {r}^(2) ) = p + q {r}^(2) \n \n Divide \: both \: sides \: by \: a - b {r}^(2) : \n = > x = \frac{p + q {r}^(2) }{a - b {r}^(2) }$