Generate Quiz 6974A60E9681E

lim x-> 0 (xp – cp/ x-c)= pcp-1 is one of the standard limits.

We must determine the value of x in this case since the domain is a set of input values for which the function is real (all real values except zero) and defined.

The values of x for which the expression under the square root is non-negative or zero must be taken into account in order to identify the domain of the function h(x)= √x2-4.

It is required that x2-4 be greater than or equal to 0:

x2-4≥0

The quadratic expression should be factored.

(x+2)(x-2)≥0

Next, identify the non-negative intervals in the expression:

x<-2: The expression is positive because both factors are negative.

-2<x<2: The expression is negative because the factor (x+2) is positive and the factor (x-2) is negative.

x>2: The expression is positive since both factors are positive.

When x -2 or x≥2, the expression x2-4 is non-negative or equal to zero, taking into account all intervals. Therefore, the domain of h(x)=x2-4 is (-∞,-2]∪[2,∞).

The domain will be {3,5,7,-9} since it contains the first element of every ordered pair, as in R={(3,0),(5,1),(7,-3),(-9,0)}.

FUNCTION COMPOSITION AND FUNCTION INVERSE:

The right response is c) 1.

We must determine the value of x for which f(x) = 1 in order to determine the value of the inverse function f^-1(1).

Given: f(x) = 1/x

We must solve the following equation in order to determine f^-1(1):

1/x = 1.

x = 1.

Therefore, the value of f^-1(1) is 1.

The following is the explanation:

The inverse function f^-1(x) is defined such that f^-1(f(x)) = x.

Since f(x) = 1/x in this instance, we must determine the value of x for which f(x) = 1.

We obtain x = 1 by solving the equation 1/x = 1.

Consequently, f^-1(1) has a value of 1.

Therefore, c) 1 is the right response.

THE LIMIT OF A FUNCTION AND LIMIT THEOREMS:

The right response is 1/2√6.

We can apply the idea of the limit of a rational function to assess this limit.

The limit of (√x – √6) / (x – 6) as x→6 can be evaluated as follows:

lim (√x – √6) / (x – 6) as x→6

= lim [(√x – √6) / (x – 6)] (Justification of the numerator)

= lim [(√x – √6) * (√x + √6) / (x – 6) * (√x + √6)]

= lim [(x – 6) / (x – 6) * (√x + √6)]

= lim 1/(√x + √6)

= 1/√6 + √6

= 1/2√6

Therefore, 1/2/√6 is the right response.

We must change g(x) into f(x) in order to find f(g(x)).

Since g(x)=3x+7 and f(x)=4x+5, let’s replace

f(g(x))=f(3x+7)

Now, replace x in f(x)=4x+5 with 3x+7:

f(g(x))=4(3x+7)+5

Give out the four:

f(g(x))=12x+28+5

Put similar terms together:

f(g(x))=12x+33

Therefore, choice (B) is the right one.

The right response is (b) 6.

To assess the specified limit:

lim (sin(6x)/x) as x->0

Since x->0, we can make use of the limit properties and the knowledge that lim (sin(x)/x) = 1.

lim (sin(6x)/x) = 6 * lim (sin(x)/x)

= 6 * 1.

= 6.

The limit evaluates to 6 as a result.

Thus, b) 6 is the right response.

We must assess the specified limit in order to resolve this issue:

lim ((1 – cos(x)) / x) as x->0

Since the limit has the form 0/0, we can evaluate it using L’Hôpital’s rule.

Utilizing the rule of L’Hôpital:

lim ((1 – cos(x)) / x) = lim (d/dx (1 – cos(x)) / d/dx (x))

= lim (sin(x) / 1)

= lim sin(x)

= 0

Thus, a) 0 is the right response.

Switching x and y and solving for the new y will allow us to determine the inverse function of:

x=y/y+5​

Now, figure out y:

x(y+5)=y

Extend and reorganize terms:

xy+5x=y

Separate y:

xy−y=−5x

y(x−1)=−5x

y = −5x / x−1

The inverse function is thus:

F-1(x)= −5x​/ x−1

Therefore, choice D is the right one.

The limit expression that is provided is:

Lim As x gets closer to 3, ((x^2 + 4x – 33) / (x – 3)

According to the question, we cannot simply enter x = 3 into the expression since doing so would cause the denominator to be zero and produce an undefined value.

The question proposes an alternative technique to get around this, which is to use the value of x after taking the derivative of the numerator and denominator independently.

The numerator’s derivative is:

d/dx = 2x + 4 (x^2 + 4x – 33)

The denominator’s derivative is:

d/dx (x – 3) = 1.

When these derivatives are substituted, we obtain:

Limit ((2x + 4) / 1) as x gets closer to 3

When we simplify this expression, we obtain:

lim (2x + 4) as x gets closer to 3.

= 2(3) + 4.

= 6 + 4.

= 10.

Consequently, 10 is the value of the specified limit.

According to the question, we are informed that this limit has a value of 14. With this knowledge, we can use the following formula to find the value of “a”:

6 + a = 14.

a = 8.

Thus, a) 8 is the ultimate response.

If a function f(x) meets the following criteria, it is considered even:

f(x) = f(-x)

This indicates that the values of the function at x and -x are identical. To put it another way, an even function’s graph is symmetric about the y-axis.An even function is not appropriately defined by the other choices. As a result, the definition of an even function, a) f(x) = f(-x), is the right response.

The parametric equations x=t2 and y=t are provided, where t fluctuates between −2 ≤ t≤ 2.

To see how x and y relate to one another, let’s remove the parameter t:

Given that x=t2 and y=t

The second equation, t=y, allows us to now express t in terms of y.

Put this in place of the first equation:

It is evident that x=(y)2 is a quadratic function.

Thus, it depicts a parabola’s equation. A parabola that opens along the positive y-axis represents the graph of this equation.

Thus, the appropriate choice is:

(b) A parabola

The correct answer is a) -sin(x).

We must examine the odd function property in order to ascertain which of the provided functions is odd:

The condition is satisfied by an odd function f(x): f(-x) = -f(x)

Let’s assess each function in relation to -x:

a) -sin(-x) = -(-sin(x)) = sin(x)

-sin(x) is an odd function since it satisfies the requirement for an odd function.

The right response is b) (2x + 3)^3.

g(x) =x3 f(x)=2x+3

gof(x)=g(f(x))=g(2x+3)=(2x+3)2.

The expression of g(x) must be entered into the function f(x) in order to find f(g(x)).

Given:

f(x) = x^2 + 1.

g(x) = 5x + 1.

Converting g(x) to f(x):

f(g(x)) = (5x + 1)^2 + 1

= 25x^2 + 10x + 1 + 1

= 25x^2 + 10x + 2.

Thus, a) 25x^2 + 10x + 2 is the right response.

The right response is (c) 2.

Information provided:

The definition of the function f(x) is f(x) = 2.

We must enter (x + 2) into the function f(x) in order to determine the value of f(x + 2).

f(x + 2) = 2.

This is due to the fact that f(x) is a constant function, meaning that, independent of the input value, it always returns the same value.

Given that f(x) = 2 for every x, f(x + 2) will likewise equal 2.

Thus, c) 2 is the right response.

If we substitute (a+b) for x in x2+3, we’ll obtain

f(a+b)=(a+b)2+3.

The expression (a+b)2 should now be expanded and made simpler:

(a+b)2=a2+2ab+b2.

Replace this with the initial expression:

f(a+b)= a2+2ab+b2+3. Or a2+b2+2ab+3

Option A is therefore the right one.

Applying the L’Hospital’s rule and taking the denominator and numerators differentiation separately, we get 3×2/2x. If you solve for x=3, you’ll get 0/o, which is indeterminate. If you solve for x=3, you’ll get 9/2.

The number of distinct function is given by 2n^2

Given that n(A) = a and n(B) = b, let A and B be two sets.

Then number of distinct relations from A to B is = 2 n(A) * n(B) = 2ab

Hence number of distinct relations from A to A is = 2 n(A) * n(A) = 2n^2

To assess the specified limit:

lim (sin(6x) / x) as x->0

The fact that lim (sin(kx) / x) = k, where k is a constant, can be applied.

Using this characteristic:

lim (sin(6x) / x) = 6

Thus, b) 6 is the right response.

The expression’s numerator contains a square root, and when determining limits, it’s frequently useful to remove square roots. To eliminate the square root, multiply the numerator and denominator by the numerator’s conjugate. This method simplifies the expression and frequently makes additional algebraic operations easier.

One of the standard limits states

Limx->∞ ex= ∞ is correct (I)

In contrast, Limx->-∞ ex= 0 rather than (II)

Substitute x with the specified limit 3, then solve as follows: (12-8)/(3+2)=4/5

FUNCTION COMPOSITION AND FUNCTION INVERSE:

The right response is c) 1.

When a function f(x) has an inverse function g(x), the reciprocal of the derivative of the original function f'(x), evaluated at the corresponding point, is equal to the derivative of the inverse function g'(x).

In mathematics, g(x) is the inverse function of f(x) if f(x) = g(x). The derivative of the inverse function g'(x) is provided by:

g'(x) = 1 / f'(g(x))

In this instance, we must evaluate the derivative of the inverse function g(x) at x = 1 in order to determine the value of g'(1).

When we replace the provided data, we obtain:

g'(1) = 1 / f'(g(1))

Given that f(x) = g(x), g(1) = 1.

Therefore, g'(1) = 1 / f'(1) = 1.

Therefore, c) 1 is the right response.

The cardinality of the set X, or the total number of elements in the set, is usually indicated by the notation ∣X∣. Considering that the number of elements in X is represented by ∿X∣. Similar to how ∣Y∣ indicates how many elements there are in Y, ∣X*Y∣ usually denotes the cardinality of the Cartesian product of X and Y, or X×Y.

The set of all possible ordered pairs (x,y) where x is from X and y is from Y is known as the Cartesian product X×Y. The formula ∣X×Y∣=∣X∣⋅∣Y∣ gives the cardinality of X×Y. Therefore, choice B is the right one.

Now let’s assess the limit:

Limθ→2π tan(θ/2)

θ/2 approaches π/4 as θ gets closer to π/2. The tangent function has a vertical asymptote at π/2 in the interval (0,π/2).

tan(θ/2)

Since θ/2​ approaches π/4​. ​, we have:

tan(θ/2) = tan(π/4)

where tan(π/4) = 1.

Consequently,

tan(θ/2), Lim θ→2π = 1

Thus, the appropriate choice is:

(a) 1

Since the given equation’s limit is zero, replacing x with zero yields an answer of 8/5, indicating that option B is the right one.

16 is the right response.

We must first determine the value of g(4) and then enter it into the function f(x) in order to find f(g(4)).

Given:

f(x) = (x + 1)^2

g(x) = x – 1

Step 1: Determine g(4)’s value.

g(4) = 4 – 1 = 3.

Step 2: Enter g(4) = 3 into the f(x) function.

f(g(4)) = f(3)

f(3) = (3 + 1)^2 = 4^2 = 16

Since the limit is zero, C is the right choice. Solve the given equation by replacing x with zero, and you will obtain 9/17.

The sequence appears to be a = 6 + (n-1)/n, where n = 1, 2, 3,…

Since the limit of (n-1)/n as n →∞ is 1 , the sequence approaches the value 6 + 1 =7

We must determine the values of a and b such that the given function f(x) is continuous in order to solve this problem.

The definition of the function f(x) is:

f(x) = ax^2 + 2x + 1, x ≤ 2

f(x) = 2x + b, x > 2

At the point where x = 2, the function values must be equal for the function to be considered continuous.

Calculating the values of the function at x = 2:

ax^2 + 2x + 1 = 2x + b

a(2)^2 + 2(2) + 1 = 2(2) + b

4a + 4 + 1 = 4 + b

4a + 5 = 4 + b

b = 5 – 4a

As a result, the following values of a and b meet the continuity condition:

a = 1 and b = 3.

The right response is (c) a = 1 b = 3.

The following characteristics must be examined in order to ascertain whether the given function, f(x) = (f(x) – f(-x))/2, is even or odd:

Even function: f(-x) = f(x)

Odd function: f(-x) = -f(x)

Let’s enter -x into the function that is provided:

f(-x) = (f(-x) – f((-)-x))/2

= (f(-x) – f(x))/2

In contrast to the original function:

f(x) = (f(x) – f(-x))/2

f(-x) = -(f(x) – f(-x))/2 is evident.

= -(f(x))/2 + f(-x)/2

= -f(x)/2 + f(x)/2

= -f(x)

This indicates that the function is odd.

Thus, b) Odd is the right response.

To evaluate the limit limx→0 e(x1/4), let’s analyze the behavior of the function as x approaches 0 .

When x gets closer to 0, x1/4 gets closer to 0, and e(x1/4) gets closer to e0, is 1.

Consequently, the limit’s proper value is:

limx→0 e(x1/4)=1

Instead of equaling zero, the limit equals one.

This is the right response. Since the logarithmic function is not defined for non-positive values of x, it has a vertical asymptote at x=0. An asymptote is a line that a curve approaches but never quite reaches in the context of a mathematical function. In particular, a vertical asymptote happens when a function’s graph gets closer to a vertical line without ever crossing it.

Since the logarithm of a non-positive number is undefined, the domain of the logarithmic function y=log(x) is limited to positive real numbers. The value of log(x) gets more negative and gets closer to negative infinity as x moves closer to 0 from the positive side. The vertical line x=0 is approached but never crossed by the graph of y=log(x). The vertical asymptote is characterized by this behavior.

In conclusion, the vertical asymptote at x=0 for y=log(x) indicates that the function is not defined for x≤0 and that the function values get more negative as x gets closer to 0 from the positive side, eventually reaching negative infinity.

The given function is f(x) = (x^2 – 4) / (x – 2)(x + 2)

We must find the values of x at which the denominator becomes zero in order to determine the vertical asymptotes.

When x = 2 and x = -2, the denominator, (x – 2)(x + 2), becomes zero.

Therefore, the vertical asymptotes occur at x = 2 and x = -2.

However, in this case, both the numerator (x^2 – 4) and the denominator (x – 2)(x + 2) become zero at x = 2 and x = -2.

There is a “hole” in the graph rather than a vertical asymptote when the numerator and denominator are both zero at the same value of x.

As a result, the given rational expression will asymptote vertically at x = -2 and have a hole at x = 2.