Answer:
True
Step-by-step explanation:
Given statement : The base of an exponential function cannot be a negative number.
We need to check whether the given statement is true or not.
The general form of an exponential function is
[tex]f(x)=ab^x[/tex]
where, a is the initial value and b is the base of exponential function.
The value of base b is always greater than 0 because
1. The terms like [tex](-3)^\frac{1}{2}[/tex] is an imaginary number and it make no sense. So, the base of an exponential function cannot be a negative number.
2. It b=0, then [tex](0)^x=0[/tex], which is a constant function. So, the base of an exponential function cannot be 0.
It means b>0.
Therefore, the given the given statement is true.
Final answer:
The base of an exponential function must be positive, as a negative base can lead to undefined values when raised to non-integer exponents. The rate of growth of an exponential function with a positive base near one can be approximated by 1+x for small x. Bases for exponential functions can often be connected to Euler's number e (approximately 2.71818).
Explanation:
The statement that the base of an exponential function cannot be a negative number is true. The base of an exponential function needs to be positive because a negative base can lead to undefined or complex values when raised to a fractional or irrational exponent. For example, the exponential function with base e (where e is the Euler's number, approximately equal to 2.71818) demonstrates natural growth, and its rate of growth for small x values approximates to 1+x. This approximation indicates how for a small change in x, the change in the value of the function is nearly proportional when the base is a positive number close to one.
Considering other bases, they can be related to base e using the identity ax = eln(a)x, where a is a positive real number, and ln(a) is the natural logarithm of a. Thus, whether for common or natural bases, the exponential function is well-defined only for positive bases.
A bicyclist travels at the rate of 8 miles in an hour. If the bicyclist continues at this rate, how many miles can the bicyclist travel in seven hours?
A. 54 miles
B. 55 miles
C. 45 miles
D. 56 miles
What is the value of 35+20/5-4*7
11
49
245
35
Hellllpp :)
A coin collector bought a set of coins for $500, and then sold the set for $850, making a profit of $50 per coin. how many coins were in the set?
For this case, the first thing we must do is calculate the total profit.
We have then:
[tex] P = 850 - 500 [/tex]
[tex] P = 350 [/tex]
Then, we look for the number of coins.
For this, we need the total profit (which we already calculate) and the amount of profit per coin.
We have then:
[tex] N = \frac{350}{50} [/tex]
[tex] N = 7 [/tex]
Answer:
The amount of coins that was on the set is:
[tex] N = 7 [/tex]
Final answer:
The coin collector bought and sold a set of coins, making a total profit of $350. Dividing this profit by the $50 profit per coin reveals that there were 7 coins in the set.
Explanation:
The coin collector made a profit of $50 per coin by selling a set of coins. To find out how many coins were in the set, we need to calculate the total profit made from selling the coins and then divide that by the profit per coin.
The total profit is the selling price minus the buying price, so $850 - $500 = $350. If the collector made $50 profit per coin, then the number of coins in the set can be found by dividing the total profit by the profit per coin: $350 / $50 per coin = 7 coins.
What is the equation of a line that has a slope of -5 and passes through the point (0, 2)?
True or false? the point at which a tangent line meets a circle is called the point of tangency
Final answer:
The statement is true; a tangent line meets a circle at the point of tangency, which is the unique point where the tangent touches the circle without intersecting it.
Explanation:
True, the point at which a tangent line meets a circle is indeed called the point of tangency. This is a fundamental concept in geometry, where a tangent line is defined as a line that touches a curve at exactly one point, without crossing through it. This distinct point is known as the point of tangency.
For example, if we have a circle and a line that touches the circle at just one point and does not intersect it at that point, that line is a tangent to the circle, and the point of contact is the point of tangency. The tangent line is always perpendicular to the radius of the circle at the point of tangency, as can also be seen from the properties of the director circle, which is the locus of points of intersection of perpendicular tangents to the curves like ellipses and hyperbolas.
In the context of a director circle, perpendicular tangents to an ellipse would intersect on this special circle. Although for this particular question, we are focusing on the tangent line to a single circle and its point of tangency. It's interesting to note that according to the properties of circles, every point on a circle allows for exactly one tangent line. Whenever such a line is drawn, it illustrates the concept of tangency.
Margie and four friends are eating lunch at a restaurant. The total bill is $52.50 before tax and tip. Margie has a coupon good for $10off their total purchase. If sales tax of 6 percent is collected on the discount amount, what is the total cost after tax?
Use the graph below to answer the following question:
What is the average rate of change from x = 1 to x = 4?
−3
−1
1
3
Answer:
Step-by-step explanation:
_______
Good evening ,
_______________
f(1)=3 and f(3) = -1 so the average rate of change from x = 1 to x = 4 is 1.
Look at the photo below for more details,
________________________________
:)
He nine ring wraiths want to fly from barad-dur to rivendell. rivendell is directly north of barad-dur. the dark tower reports that the wind is coming from the west at 68 miles per hour. in order to travel in a straight line, the ring wraiths decide to head northwest. at what speed should they fly (omit units)?
This problem is better understood with a given figure. Assuming that the flight is in a perfect northwest direction such that the angle is 45°, therefore I believe I have the correct figure to simulate the situation (see attached).
Now we are asked to find for the value of the hypotenuse (flight speed) given the angle and the side opposite to the angle. In this case, we use the sin function:
sin θ = opposite side / hypotenuse
sin 45 = 68 miles per hr / flight
flight = 68 miles per hr / sin 45
flight = 96.17 miles per hr
During a recent snowstorm in red hook ny Jaime notice that there were 4 inches of snow on the ground at 3:00p.m and there were 6 inches of snow on the ground at 7:00p.m. If she were to graph these data what does the slope of the line connecting these two point represent in the context of this problem
Jill spend $15 on diapers. If they cost $5 a package, how many packages did she buy ?
If isosceles triangle ABC has a 130° angle at vertex B, which statement must be true?
m∠A = 15° and m∠C = 35°
m∠A + m∠B = 155°
m∠A + m∠C = 60°
m∠A = 20° and m∠C = 30°.
You are saving money to buy a new game system you received $50 as a birthday gift from your grandparents. You want to save $25 a week from mowing lawns.
Determine the distance between a(-1, 2,3) and b(4,-5, 6) to the nearest tenth
Find the prime factors of each of the following numbers. a. 25 b. 38 c. 36 d. 285 e. 650 f. 1,740
Answer:
The prime factors are:
a. 5 and 5 (5 × 5 = 25)
b. 2 and 19 (2 × 19 = 38)
c. 2, 2, 3, and 3 (2 × 2 × 3 × 3 = 36)
d. 3, 5, and 19 (3 × 5× 19 = 285)
e. 2, 5, 5, and 13 (2 × 5 × 5 × 13 = 650)
f. 2, 2, 3, 5, and 29 (2 × 2 × 3 × 5 × 29 = 1,740)
Step-by-step explanation:
straight from Penn
Find all solutions to the equation in the interval [0, 2π). cos 2x - cos x = 0
Answer:
The other answer is correct here except it is missing the answer 4pi/3
other wise its correct. The answers are 0, 2pi/3, and 4pi/3
Step-by-step explanation:
see previous answer
What is the height, h, of one of the triangles?
Answer:
8ft
Step-by-step explanation:
The base of a cone can be
A. a triangle.
B. a pentagon.
C. a square.
D. an ellipse.
Double vision can be the result of _________________.
Wearing glasses that are too strong
Looking directly into bright headlights Passing other cars
Eye muscles relaxed from fatigue, drugs, or alcohol
Rapid eye blinking
Final answer:
Double vision can be caused by wearing glasses that are too strong, eye muscles being relaxed from fatigue, drugs, or alcohol, or by rapid eye blinking. Nearsightedness and farsightedness are common vision defects related to the eye's focusing ability but do not usually cause double vision, which can also be a result of misalignment.
Explanation:
Double vision can be the result of several factors including wearing glasses that are too strong, eye muscles relaxed from fatigue, drugs, or alcohol, or rapid eye blinking. It is not typically caused by looking directly into bright headlights or the action of passing other cars, which are more likely to cause temporary glare or flash blindness rather than persistent double vision.
Common vision defects such as nearsightedness (myopia) and farsightedness (hyperopia) are related to the focusing ability of the eyes. Myopia occurs when the eye over-converges the rays from a distant object in front of the retina, causing distant objects to appear blurry. Hyperopia, on the other hand, happens when the eye does not converge rays from a close object sufficiently, making close objects appear blurry. Misalignment can also cause double vision, and this can be due to a variety of factors including strabismus or nerve damage. Correction of vision defects often involves the use of glasses or contact lenses with the appropriate lens power or surgery for more severe cases.
Find the missing factor x(y+6)-5(y+6)( ) (y+6)
We are given the equation:
x (y + 6) – 5 (y + 6) ---> 1
We are to look for the missing factor in the equation: (_) (y + 6) from the given values.
Knowing the rules of the polynomial multiplicity, we can have an idea that the given equation can also be written in the form of:
(x – 5) (y + 6) ---> 2
wherein the whole factor (y + 6) was simply distributed to the values in the factor (x – 5). To prove that equations 1 and 2 are equal:
x (y + 6) – 5 (y + 6) = (x – 5) (y + 6)
Expanding:
x y + 6 x - 5 y – 30 = x y + 6 x – 5 y – 30 (EQUAL!)
Therefore our missing factor is:
(x – 5)
Fill in the missing term in the equation. x/x^2-4x+4-x/x^2-3x+2=?/(x-2)^2(x-1)
Answer:
The missing term is x.
Step-by-step explanation:
Given expression is,
[tex]\frac{x}{x^2-4x+4}-\frac{x}{x^2-3x+2}------(1)[/tex]
Since, [tex]x^2-4x+4=(x)^2-2\times (2x)+(2)^2[/tex]
[tex]\implies x^2-4x+4=(x-2)^2------(2)[/tex] ( (a-b)² = a² - 2ab + b² )
Now, [tex]x^2-3x+2=x^2-2x-x+2[/tex] ( By middle term splitting )
[tex]=x(x-2)-1(x-2)[/tex]
[tex]\implies x^2-3x+2=(x-1)(x-2)------(3)[/tex]
From equation (1), (2) and (3),
[tex]\frac{x}{x^2-4x+4}-\frac{x}{x^2-3x+2}=\frac{x}{(x-2)^2}-\frac{x}{(x-2)(x-1)}[/tex]
[tex]=\frac{x(x-1)-x(x-2)}{(x-2)^2(x-1)}[/tex]
[tex]=\frac{x(x-1-x+2)}{(x-2)^2(x-1)}[/tex]
[tex]=\frac{x(1)}{(x-2)^2(x-1)}[/tex]
[tex]=\frac{x}{(x-2)^2(x-1)}[/tex]
Hence, the missing term in the equation is x.
What is the polygon being rotated to create the brown quilt block pattern?
Answer with explanation:
The pre image which is in blue color is parallelogram.
There are four blue parallelograms here,congruent to each other,each one of them when rotated by certain angle ,gives parallelogram in brown color.
When we rotate a polygon,after rotation neither the shape nor the size of polygon changes.The two shapes are congruent to each other.
→The Polygon being rotated to create the brown quilt block pattern is
Option C: Parallelogram
A circle has a diameter of 13 centimeters and a central angle EOG that measures 280 degrees. What is the length of the intercepted arc EG? Use 3.14 for pi and round your answer to the nearest tenth. A) 10.1 cm B) 15.9 cm C) 31.7 cm D) 63.5 cm
Which polynomial is a perfect square trinomial?
A. 36x²-18x-9
B. 4a²-28a-49
C. 25b²-40b+16
D. 4x²-6x+9
37. A juice pitcher holds 1.5 gallons of liquid. How many 8-ounce glasses of juice can be poured from a full pitcher? (1 gallon 128 ounces) Explain your answer by writing or describing the steps you used to solve the problem.
1 gallon = 128 ounces
multiply 128 by 1.5 gallons:
1.5 x 128 = 192 ounces total
divide total ounces by 8 to get number of glasses:
192/8 = 24
you can pour 24 eight ounce glasses
25 POINTS PLZ HELP ASAP!!!
Match the functions with their inverse functions.
We know that when we are given a inverse function as: [tex]f^{-1}(x)[/tex]
Then we find the function by the method:
Put [tex]f^{-1}(x)=y[/tex]
Then we switch the places of x and y and solve for y.
1)
[tex]f^{-1}(x)=\dfrac{2x-3}{5}[/tex]
Hence, we find the function as follows:
[tex]\dfrac{2x-3}{5}=y[/tex]
Then we switch for x and y
[tex]\dfrac{2y-3}{5}=x\\\\\\i.e.\\\\2y-3=5x\\\\\\i.e.\\\\\\2y=5x+3\\\\\\i.e.\\\\\\y=\dfrac{5x+3}{2}[/tex]
Hence, inverse function is:
[tex]f(x)=\dfrac{5x+3}{2}[/tex]
2)
[tex]f^{-1}(x)=\dfrac{x+8}{2}[/tex]
Hence, we find the function as follows:
[tex]\dfrac{x+8}{2}=y[/tex]
Then we switch for x and y
[tex]\dfrac{y+8}{2}=x[/tex]
[tex]y=2x-8[/tex]
Hence, inverse function is:
[tex]f(x)=2x-8[/tex]
3)
[tex]f^{-1}(x)=\dfrac{x+2}{7}[/tex]
Hence, we find the function as follows:
[tex]\dfrac{x+2}{7}=y[/tex]
Then we switch for x and y
[tex]\dfrac{y+2}{7}=x[/tex]
[tex]y=7x-2[/tex]
Hence, inverse function is:
[tex]f(x)=7x-2[/tex]
4)
[tex]f^{-1}(x)=\dfrac{1-2x}{x}[/tex]
Hence, we find the function as follows:
[tex]\dfrac{1-2x}{x}=y[/tex]
Then we switch for x and y
[tex]\dfrac{1-2y}{y}=x[/tex]
i.e.
[tex]1-2y=xy\\\\xy+2y=1\\\\i.e.\\\\y(x+2)=1\\\\y=\dfrac{1}{x+2}[/tex]
Hence, inverse function is:
[tex]f(x)=\dfrac{1}{x+2}[/tex]
What is the result of isolating y2 in the equation below? 9x2 + 7y2 = 42
Isolating a variable involves changing the subject of the equation.
The result of isolating y^2 is: [tex]\mathbf{y^2 = \frac{42 - 9x^2}7}[/tex]
The equation is given as:
[tex]\mathbf{9x^2 + 7y^2 = 42}[/tex]
Subtract 9x^2 from both sides
[tex]\mathbf{7y^2 = 42 - 9x^2}[/tex]
Divide through by 7
[tex]\mathbf{y^2 = \frac{42 - 9x^2}7}[/tex]
Hence, the result of isolating y^2 is: [tex]\mathbf{y^2 = \frac{42 - 9x^2}7}[/tex]
Read more about subject of formulas at:
https://brainly.com/question/21866313
PLEASE HELP ASAP 20 POINTS PLUS BRAINIEST!
Find the greatest possible error for each measurement.
3 ft
a.
1 ft
b.
1/2 ft
c.
1/4 c
d.
1/6 c
the greatest possible error when measuring in feet is 1/2 foot
answer is B
what are the roots of the polynomial equation x3-7x=6x-12? Use a graphing calculator and a system of equations.
The roots of the polynomial equation x³ - 7x = 6x - 12 is x = 1, 3, -4 and graph is shown in the picture.
What is polynomial?Polynomial is the combination of variables and constants systematically with "n" number of power in ascending or descending order.
[tex]\rm a_1x+a_2x^2+a_3x^3+a_4x^4..........a_nx^n[/tex]
We have a polynomial equation:
x³ - 7x = 6x - 12
[tex]\rm x^3-7x+12=6x[/tex]
x³ - 13x + 2 = 0
(x - 1)(x -3)(x + 4) = 0
x = 1, 3, -4
Thus, the roots of the polynomial equation x³ - 7x = 6x - 12 is x = 1, 3, -4 and graph is shown in the picture.
Learn more about Polynomial here:
brainly.com/question/17822016
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Complete the square to solve the equation below.
x²+x =7/4
A.)x=3, x= -1
B.)x= -2+ √2, x= -2- √2
C.) x= -1/2+√2, x= -1/2- √2
D.) x=1+ √-2, x= 1- √2
Answer: The correct answer is C
Step-by-step explanation: APPEX learning quiz
Explain how to write and evaluate an algebraic expression.
one more than the product of a number and seven decreased by five; evaluate when n=6
Answer:
Replace critical words with variables, numbers, and operations. More than is addition, the product is multiplication and decreased by is subtraction. The constants are 1, 7, and 5. The expression is 1 + 7n – 5. To evaluate the expression, substitute 6 for the variable and then simplify the order of operations. The value when n = 6 is 38.Step-by-step explanation: