Functions Questions and Answers

Math
FunctionsAnalyze the polynomial function f(x)= x^3 + x^2 - 6x
(a) Determine the end behavior of the graph of the function.

Math
FunctionsIf f(x) = (x +1)/x-1, find the domain of f.
a)all real number greater than 1
b)all real numbers except 1 and -1
c)all real numbers less than -1
d)all real numbers
e) all real numbers except 1

Math
FunctionsIf f(x) = -4|3x - 2| and g(x) = -20, solve the following.
(a) f(x) = g(x)
(b) f(x) >g(x)
(c) f(x) = g(x)
(a) Solve f(x) = g(x). Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A. The solution set is {□ }.
(Simplify your answer. Use a comma to separate answers as needed.)
B. The solution set is {x| □} .
(Simplify your answer. Type a compound inequality.)
C. The solution set is {x|x≤□ or x≥□}.
(Simplify your answer. Type inequalities.)
D. The solution set is the empty set.
![The function below gives the cost in dollars to manufacture x items:
C (x) = 10,000 + 5x-x^2/10,000
Find the average cost per item over the interval [1,000, 1,010].](https://media.kunduz.com/media/sug-question/raw/76848710-1658847224.8190172.jpeg?w=256)
Math
FunctionsThe function below gives the cost in dollars to manufacture x items:
C (x) = 10,000 + 5x-x^2/10,000
Find the average cost per item over the interval [1,000, 1,010].

Math
FunctionsThe plane area is bounded by the functions y=x^2 and y=4x - x^2 Find the volume generated when the plane area is revolved about the line y=6

Math
FunctionsLet f(x) =– 4x² +5/1 +x²
The graph of f is given below. Use the graph of f to determine the given limits. Enter DNE if a limit fails to exist, except in case of an infinite limit. If an infinite limit exists, enter ∞, or -∞, as appropriate.
lim f(x)=________
x->-∞
lim f(x)=________
x->∞

Math
FunctionsFor the function f(x) = 6x2 + 2x – 4, evaluate and fully simplify each of the following.
f(x+h) =
f(x+h)-f(x)/h =

Math
FunctionsGiven that there exists a surjective function f : S -> T, which of the following substitutions for the sets S and T are possible? Select only one answer.
(a) S = {1,2,3}. T= R.
(b) S = {x ∈ R : x² - 4 = 0}, T = {5, π, -10}
(C) S = {x ∈ R : (x² - 1)(x² - 4)=0}, T = {5,π,-10}
(d) S = {1}, T = {x ∈ R : eˣ = 0}.
(e) S = {n ∈ N : n ≤ 2022}, T={z ∈ Z : |z| ≤ 202}.


Math
FunctionsAlyssa was given a gift card for a coffee a shop. Each morning, Alyssa uses the card to buy one cup of coffee. Let A represent the amount money remaining on the card after buying x cups of coffee. The table
below has select values showing the linear relationship between x and A. Determine the original amount of money on the gift card.
X A
2 15.00
4 10.00
6 5.00

Math
FunctionsThe function f(t) = 8000(0.15)^24t represents the change in a quantity over t days. What does the constant 0.15 reveal about the rate of change of the quantity?
The function is____exponentially at a rate of____ % every

Math
FunctionsFind the domain of the function f(x) =(x^2 +2x +1)/x+1
a)Set of negative real numbers
b)Set of real numbers except -1
c)Set of real numbers except 1

Math
FunctionsSolve by graphing:
-x+y=-2
-2x+2y=-4
Select the correct answer below:
a)Infinite solutions in the form (x,-2-2)
b)Infinite solutions in the form (x,-x+2)
c)Infinite solutions in the form (x,x - 2)
d) Infinite solutions in the form (x, x+2)
e)One solution: (1, -1)
f)No solution:

Math
FunctionsUse a graphing calculator to solve the nonlinear system.
y =e^(X+1)
X+y =1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A). The solution set is (Type an ordered pair. Use a comma to separate answers as needed. Type an integer or decimal rounded to the nearest hundred)
B). There are infinitely many solutions.
C). There is no solution.

Math
FunctionsConsider the function f : N→P(N) given by
f(x) = {d ∈ N: d|x}.
(a) Write out all of the elements of the set f(10) = {d ∈ N: d|x}.
(b) Is f one-to-one? Prove your answer.

Math
FunctionsFor the given function: f(x) =x-5/(x^2 -25)
A). State the domain of the given function.
B). Find the value of limx→5 f(x), if it exists. Justify your answer.
C). Find the value of limx,→∞ f(x), if it exists. Justify your answer.

Math
FunctionsWrite the equation of the exponential function represented by the table. (round each to the nearest thousandths).
X 5 3 1 -1
Y 32 16 8 4

Math
FunctionsGiven that
p₁ (t) = 2t² + 2t - 1 and p₂ (t) = –t² – 2t + 1,
which of the following polynomials in P₂ does not belong to span{p₁ , p₂}? Select one: :
(A) p(t) = t² - 2t + 1
(B) p(t) = 8t² + 10t -5
(C) None of these.
(D) p(t) = -7t² - 10t + 5
(E) 4t² + 2t - 1


Math
FunctionsFunction r(t) gives the amount of rainfall accumulated in inches, when it is t hours after the rain started. Which statement represents the meaning of the equation r(5) = 2.5 in this situation?
2.5 hours after the rain started, it has rained 5 inches
5 days after the rain started, it has rained 2.5 inches
5 hours after the rain started, it has rained 2.5 inches
It has rained 5 inches for 2.5 hours

Math
FunctionsLet f(x) = (1 + 4x^2)(x - x^2).
a)Find the derivative by using the Product Rule.
f'(x) =_________
b)Find the derivative by multiplying first.
f'(x) =_________
c)Do your answers agree?
1)Yes
2)No


Math
FunctionsOreo was asked to find the end behavior of f(x) = 3x² +2 -7x^5+ 11. In which step did she make an error? Explain what her error was. Type your answer below.
Step 1: She found the leading coefficient to be 3x^2
Step 2: She found the coefficient to be positive 3, so the function is increasing from left to right
Step 3: She found the exponent to be even, so the ends are pointing in the same direction
Step 4: Her answer was_______

Math
FunctionsFor the function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.
f(x) = 2/7x+4
Find the formula for f⁻¹(x), the inverse of f(x) = 2/7x+4 if it exists. Select the correct choice below and, if necessary, fill in any answer boxes within your choice.
A. The function is one-to-one and f⁻¹(x) = ▢.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
B. The function is not one-to-one and there is no inverse function.

Math
FunctionsEvaluate f(3).
f(x) = -x+3 if x < 2
2x-3 if x ≥ 2
A) f(3) = 0
B) f(3) = 6
C) undefined
D) f(3) = 3

Math
FunctionsSolve the equation by factoring: (x + 4)(x + 5) = 11x + 68
x=
Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and-2/3 as your answers, then enter 4,-2/3 in the box.
![Consider the function f(x) = 3x^5/3 + 60x^2/3.
(a) Show that f'(x) = {[5(x + 8)] / ∛x}. Then find and classify all local extrema of f.
b) Find the values of the global extrema of f on the domain [-1,1].
c) Justify briefly, but precisely, why f has global extrema on the interval [-1, 1]](https://media.kunduz.com/media/sug-question/raw/77914624-1658831057.2986515.jpeg?w=256)
Math
FunctionsConsider the function f(x) = 3x^5/3 + 60x^2/3.
(a) Show that f'(x) = {[5(x + 8)] / ∛x}. Then find and classify all local extrema of f.
b) Find the values of the global extrema of f on the domain [-1,1].
c) Justify briefly, but precisely, why f has global extrema on the interval [-1, 1]

Math
FunctionsFor the given functions f and g, find the requested function and state its domain.
f(x) = 9x - 2; g(x) = 7x - 8.
Find f-g.
(f - g)(x) = 2x + 6; all real numbers
(f - g)(x) = 2x - 10; {x|x ≠ 5}
(f - g)(x) = -2x - 6; all real numbers
(f - g)(x) = 16x - 10; {x|x ≠ 1}

Math
FunctionsIsaac invested $1,800 in an account paying an interest rate of 4.7% compounded
continuously. Assuming no deposits or withdrawals are made, how much money, to the nearest hundred dollars, would be in the account after 15 years?

Math
FunctionsLet T: P₃->P₃, be the linear transformation such that
T(2x²) = -2x² + 3x. T(-0.5x - 2) = -4x² - 4x -2, T(5x² + 1) = 2x - 1.
Find T(1), T(x), T(x²), and T(ax² + bx + c), where a, b, and c are arbitrary real numbers.
T(1) =
T(x) =
T(ax² + bx + C) =

Math
FunctionsCalculate the average rate of change of the following functions.
1. f(x) = x² over the interval 2 ≤ x ≤ 3
2. h(x) = 7 - 3x² over the interval 2 ≤ x ≤ 5
Criteria for Success
The difference f(x2)-f(x1) is calculated.
The difference (x2) - (x1) is calculated.
Final answer is expressed with correct units.

Math
FunctionsThe equation below shows the cost C in dollars, of a small company to produce units in one week. C(x) = 500 + 60x +0.5x^2? How much money will it cost to produce 28 units in one week?

Math
FunctionsGraph an exponential that passes through the point (2,6) and has growth factor 3. Plot and label at least 3 other points (include an intercept).

Math
FunctionsA certain radioactive element has a half-life of 35 days. If you had a 100 grams of this element, the mass m of the element after t 35- day intervals is represented by m= 100(0.5)t. Find the approximate daily decay rate of this element.
a. 98.0%
b. 17.5%
c. 2.0%
d. 1.4%

Math
FunctionsFind the domain of the composite function f₀g.
f(x) = x + 4; g(x) = 5/x+6
{x | x is any real number}
{x | x ≠ -6}
{x| x ≠ -10)
{x | x≠ -6, x ≠ -4}

Math
FunctionsLet f(x)=x²-9 and g(x) = 6 -x. Perform the composition or operation indicated.
(fg)(-1)

Math
Functions(b) Find the coordinates of the vertex.
(c) Find the intercept(s).
For both the x- and y-intercept(s), make sure to do the following. If there is more than one, separate them with commas. .If there are none, select "None".
x-intercept(s):
y-intercept(s):
(d) Find the equation of the axis of symmetry. equation of axis of symmetry:


Math
FunctionsWrite the equation in standard form and enter it below. Do not enter blank spaces in your answer.
y=3/4 x+2
Standard form of equation:

Math
FunctionsLet f(x) = 3x² - 8x + 5 and g(x) = x² + 16. Find ƒ + g, ƒ − g, f · g, and Simplify your answers.
1. f + g =
2. f-g=
3. f . g =
4. f/g =

Math
FunctionsThe function P(t) = 37.25e0.0096t. approximates the population, in millions, of California t years since 2010. Assuming this trend continues, approximate the number of years it will take California's population to reach 50 million. Round your answer to two decimal places.

Math
FunctionsIf the point (5,-2) is on the graph of a function that is shifted to the right by 3, what point is on the new graph?
(5, 1)
(8.1)
(8,-2)
(2,-2)

Math
FunctionsSolve each equation
a) Solve x2/3 - 5x¹/3 +6=0
b) Solve p4-3p2-4 = 0
c) Solve x² - 4x-1-5 = 0

Math
FunctionsWhich of the following represents the rectangular equation x² + y² + 5x - 3y = 0 in polar form?
r= -5sin θ + 3cos θ
r= -3sin θ+ 5cos θ
r=3sin θ- 5cos θ
r= 5sin θ- 3cos θ



Math
FunctionsA polar curve is represented by the equation r₁ = 2 + 3cosθ.
Part A: What type of limaçon is this curve? Justify your answer using the constants in the equation.
Part B: Is the curve symmetrical to the polar axis or the line θ = π/2? Justify your answer algebraically.


Math
FunctionsSuppose f(q) = C = 200+ 0.1q gives the cost, C, in dollars to manufacture q kg of a chemical.
Find and interpret f-¹(C).

Math
Functionsa) Find and classify the critical point(s).
b) Find the interval(s) where f(x) is increasing.
c) Find the interval(s) where f(x) is decreasing.
1. f(x)=x²-x-1
2. f(x)=2x^4 - 4x² +1
3. f(x)=xe^1/x