Functions Questions and Answers

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

If 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

If 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].

The 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

Let 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->∞

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

Given 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}.

Graph the linear equation; f(x) =8x

Alyssa 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

The 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

Find 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

Solve 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:

Use 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.

Consider 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.

For 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.

Write 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

Given 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

Evaluate the expression ( x⁴+7x³-2)/(y⁴+2) , when x = 4 and y=-2. -39 39 38

Function 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

Let 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

Graph the inequality. 5x - 6y >= 0 Use the graphing tool to graph the inequality.

Oreo 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_______

For 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.

Evaluate 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

Solve 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]

For 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}

Isaac 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?

Let 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) =

Calculate 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.

The 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?

Graph 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).

A 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%

Find 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}

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

(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:

Solve by any method. 6x - 8y = -0.6 3x + 2y = 0.9

Write 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:

Let 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 =

The 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.

If 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)

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

Which 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

Find the inverse function of y = g(x) = 5x³ + 1 g-¹(y) =

Check that the functions f(x) = e2x and g(x)=In(x)/2 are inverses using algebra.

A 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.

Decompose the function y = 2x+1 into u(v(x)). If we assume u(x) = 2x, what is v(x)?

Suppose 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).

a) 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