Functions Questions and Answers

Find the gradient of the function at the given point. f(x, y, z)=√x² + y² + z² , (5, 2, 9) Find the maximum value of the directional derivative at the given point.

Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x)=x² +8x+15

Find a 4th degree polynomial function with real coefficients satisfying the given conditions. n = 4; 21, 7, and -7 are zeros; leading coefficient is 1 f(x)=x^4-45x^2-196 f(x) = x^4 + 4x^3-45x²-196 f(x) = x^4 + 4x^2-7x - 196 f(x) = x^4 + 4x²-196

The value of a family's home is given by f(x) = 130 000 (1.06)*, where x is the number of years after the family purchases the house for $130 000. What is the best estimate for the instantaneous rate of change in the value of the home when the family has owned it for 5 years? a. $8000/year b. $10 000/year c.$12 000/year d.$14 000/year

For the following equation of a function, (a) find the zero of the function, (b) find the x-intercept of the graph of the function, and (c) solve the equation f(x) = 0. f(x) = 160+32x (a) What is the zero of f(x)? (b) What is the x-intercept? (c) What is the solution to f(x)=0?

First-class postage is $0.32 for the first ounce (or any fraction thereof) and $0.24 for each additional ounce (or fraction thereof). Let C(x) represent the postage for a letter weighing x oz. Use this information to answer the questions. A. C(3) = B. The solution is undefined. e) Find all values on the interval (0,4) where the function is discontinuous.

Evaluate the following expressions for the given values. 9x4 for x = -8 2-7x for x =-3 1/2bh for b= 8, h = 10 4x²+2x-9 for x = 7

For the function f(x) = (x+3)², find the domain on which the function is one-to-one and non-decreasing, and then find the inverse of the function on this domain. Show your step-by-step process.

Consider the function H defined by H(x) = −6x – 5. Evaluate the following: a. H (5) = b. H(-7)=

Consider the rational function given by f(x) = 3/x-1. Part B: Determine where any zeros, holes, or vertical asymptotes are located. For these blanks, answer with a number, a list of numbers separated by commas ordered from least to greatest, or if no value applies, "n/a" without quotes. What values of x make the numerator equal to zero? The numerator becomes zero at x= What values of a make the denominator equal to zero? The denominator becomes zero at x = Where are the zeros, if any, for this graph? The zero(s) of the graph of this function occur(s) at the values x = Where are the vertical asymptotes, if any, for this graph? This graph has vertical asymptotes at the values x = Where are the "holes" (also called "point..." or "removable... discontinuities) for this graph? The graph of this function has-holes at the values x =

The cost of a Frigbox refrigerator is $800, and depreciates $40 each year. The cost of an Arctic Air refrigerator is $1080, and it depreciates $110 per year. (a) Find an equation for the value of the Frigbox, F, t years after it is purchased. (b) Find an equation for the value of the Arctic Air, A, t years after it is purchased. (c) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value?

4) What is the effect on the graph of the function f(x) = 7x³ when f(x) is replaced with f(x - 5)? A) translate vertically 5 units up B) translate vertically 5 units down C) translate horizontally 5 units left D) translate horizontally 5 units right

The gross tonnage G is a standardized measure of a ship's capacity. It is calculated in terms of the volume V, in cubic meters, of the ship. There are no units associated with gross tonnage. It is calculated using the formula G= V(0.2 + 0.02 log V). In this exercise round your answers to the nearest whole number. (a) Find the gross tonnage of a ship with a volume of 15,000 cubic meters. (b) Use the crossing-graphs method to find the volume of a ship with a gross tonnage of 7000.

The function C(t) = Co(1 + r)^t models the rise in the cost of a product that has a cost of Co today, subject to an average yearly inflation rate of r for t years. If the average annual rate of inflation over the next 11 years is assumed to be 3.5 %, what will the inflation-adjusted cost of a $31,000 motorcycle be in 11 years? Round to two decimal places.

The speaker constructs a formula for the distance from the origin to a particular point on the graph of the function y=x²-8. What is that distance formula? The distance formula is.

A laptop computer was purchased for $1450. Each year since, the resale value has decreased by 22%. Lett be the number of years since the purchase. Let y be the resale value of the laptop computer, in dollars. Write an exponential function showing the relationship between y and t.

Given the function f(x) =6x^4 - x³ + x² - 42x + 7. There are real zeros There are x intercepts There are turning points

A regression equation of y = 0.923x + 37.047 is found that gives the yield of bushels of oats per acre () given an amount of rainfall (X). Using the model, predict the yield when there is 7 inches of rain. Select the correct answer below: 44.9 bushels 43.5 bushels 39.1 bushels 37.8 bushels

A particular curve is represented parametrically by x = 3t+1, y = t²-4, for t ∈ [-1,3]. (a) What is the corresponding Cartesian equation for this curve (the equation in x and y only)? y = (b) Give the smallest and largest values of x taken on by this curve (c) Give the smallest and largest values of y taken on by this curve Note: You can earn partial credit on this problem.

In a certain country, income tax is assessed by people not having to pay a tax if they make $15,000 per year or less, people having to pay a 12% tax if they make over $15,000 but no more than $30,000, and people having to pay 18% tax if they make more than $30,000 per year. Find the piecewise function that represents this scenario.

10. Marty was renting a boat and wanted to get the best plan. Rental plan A charges a fee of $50 plus $25 per hour. Plan B charges a fee of $1 plus $35 per hour. Using systems of equations, answer the following questions. Show all your work. Part A: Write a system of equations the represent the situation. (Be sure to identify the variables.) Part B: When will the plans cost the same amount? Explain how you found your solution.

Kristin's grandparents started a savings account for her when she was born. They invested $500 in an account that pays 8% interest compounded annually. a. Write an equation to model the amount of money in the account on Kristin's x^th birthday. b. How much money is in the account on Kristin's 16th birthday? c. What are the domain and range of the equation that you wrote in part (a)?

Which of the following is true of an even function? It is symmetric about the x-axis The function f (x) = 0 is and odd function but not an even function It can't have an inverse because it is not one-to-one The sum and product of 2 even function is odd (x + 1)² Is an even function because of the exponent 2.

Rewrite the exponential function h(z) = 5/3 2^x3^x+1 in the form h(z) = A· B^x A= B = The y-intercept is the point(,)

Find the formula for an exponential function of the form f(x) = A. B^x that passes through the points (2, 125) and (4, 3125). f(x) =

If f(x) is a rational function, what is one way to find the solution to f(x)=4 graphically? Choose the correct answer below. A. Find any x-intercepts. The x-coordinates of these intercepts are the solutions. B. Graph y=f(x). Find the x-value of 4 on the x-axis and move up or down to a point on the graph. The y-coordinate of this point is the solution. C. Graph y=f(x) and y=4 in the same xy-plane and find any points of intersection. The x-coordinates of these points are the solutions.

If the price of a camping vacation can be expressed as a function of the number of nights, what does the inverse function represent? Number of nights as a function of the price per night Price of the vacation as a function of the price per night Price of the vacation as a function of the number of nights Number of nights as a function of the price of the vacation

Shana is climbing a mountain at a constant pace. She rises in elevation by 17.5 feet per minute. After hiking for 30 minutes, a GPS app tells her that her elevation is 2,165 feet above sea level. Write a linear function for Shana's elevation, E, above sea level as a function of the time, t, she has been climbing in minutes.

The x-intercepts of the quadratic equation y = -x²+bx+c are (1-√18,0) and (1+√18,0). (a) Graph the quadratic equation using the information above. (b) Use your graph from (a) to find all solutions of -x²+bx+c <0 and graph your solution set on the number line below.

6Element X decays radioactively with a half life of 5 minutes. If there are 390 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 170 grams? Given the equation y = a(0.5)^t/h with y = 170, a = 390, h = 5 Find t.

A museum charges $7 per person for a guided tour with a group of 1 to 11 people or a fixed $84 fee for a group of 12 or more people. Write a function relating the number of people, n, to the cost, C.

How many real roots does the quadratic function h(x) = 4x² - 12x +9 have? 0 1 2

Domain and Range The function f(x) is defined as a set or ordered pairs. Using that set of data, identify the Domain and Range of f(x). Write your answer as an ordered list enclosed in curly brackets. Question Hal f(x) = {(-11, 7), (-1, 51), (13, 90), (16, 58), (40, 3), (49, 91)} Domain: Range:

Which of the following are one-to-one functions? - S = {(-5, 8), (-3,-7), (9,- 6), (1,7), (8, — 3), (1,6)} OR={(-5, 4), (2, 1), (5, 4), (9, 7), (10, 13), (14, 15)} OF = {(-4, 5), (0, 0), (1, 3), (4, 5), (8, 7), (13, 16)} - OG = {(3, 1), (- 4, 1), (4, 2), (-5, 1), (7, 1), ( – 3, — 2)} OM= {(-5, - 3), (3, 1), (3, 4), (6, 9), (7, — 6), (10, 14)} - - -

A cooler has a temperature of 32 degrees Fahrenheit. A bottled drink is placed in the cooler with an initial temperature of 70 degrees Fahrenheit. The function f(t) = Ce^(-kt) + 32 represents the situation, where t is time in minutes, C is a constant, and k is a constant. After 3 minutes the bottle has a temperature of 42 degrees. What is the approximate value of k? Select the correct answer below: 0.445 0.465 0.497 0.512 0.541

Jocelyn is going to the amusement park, where she has to pay a set price of admission and another price for tickets to go on each of the rides. The total amount of money Jocelyn will spend is given by the equation y = 3.5x + 28, where y represents the total amount of money, in dollars and cents and a represents the number of rides Jocelyn goes on. What could the number 28 represent in the equation? The total amount of money Jocelyn will spend if she goes on 1 ride. The change in the total amount of money for every one additional ride she goes on. The total amount of money Jocelyn will spend if she goes on 100 rides. The cost to get into the amusement park.

Transform the given function f(x) as described and write the resulting function as an equation. 3) f(x)= x³ expand vertically by a factor of 3 reflect across the x-axis translate left 2 units translate up 3 units g(x) = -3(x + 2)³ + 3

What effect does replacing x with x + 6 have on the graph for the function f(x)? f(x)= ⎮x+3⎮-2 The graph is shifted left 6 units. The graph is shifted right 6 units. The graph is shifted down 6 units. The graph is shifted up 6 units.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function. State the domain of f. f(x)= 1-2x/4x+1 Identify any vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is one vertical asymptote. Its equation is (Type an equation. Use integers or fractions for any numbers in the equation.) B. There are two vertical asymptotes. The equation of the leftmost one is (Type equations. Use integers or fractions for any numbers in the equations.) C. There are no vertical asymptotes. and the equation of the rightmost one is

At the beginning of a population study, a city had 380,000 people. Each year since, the population has grown by 4.1%. Let t be the number of years since start of the study. Let y be the city's population. Write an exponential function showing the relationship between y and t.

Graph the function, f(x)=1/4 |2x| - 1 Use the Ray tool to graph the function as two rays with endpoints at the vertex of the graph.

Tablet Computers Annual sales (in millions of units) of a certain brand of tablet computers are expected to grow in accordance with the function f(t) = 0.18t² + 0.16t+ 2.64 (0 ≤ t ≤ 10) per year, where t is measured in years, with t = 0 corresponding to 1997. How many tablet computers will be sold over the 10 year period between the beginning of 1997 and the end of 2006? (Round your answer to two decimal places.)

Medical researchers measured the populations of bacteria in a petri dish after treatment with the new antibiotic as well as in a petri dish that was untreated. The graph above plots the populations of bacteria in both dishes. Which of the following expressions shows the difference in population between the treated petri dish and the control dish t hours after treatment? A) 5,000(0.95) - 4,000(0.9) B) 4,000(0.95)t - 5,000(0.9)t C) 5,000(0.9) - 4,000(0.95)' D) 5,000(1.1) - 4,000(1.05)'

Charlie worked for a stone setting company this past summer. He earned $50 just for coming to work and an additional $30 every hour for each hour he worked. The linear function E(t) = 50+ 30t represents Charlie's earnings E as a function of time worked t (in hours). Find E(6) + E(2)

First find f+g, f-g, fg and f/g. Then determine the domain for each function. f(x) = 4x-8, g(x)=x+3 (f+g)(x) = (Simplify your answer.)

6x + 7 X-2' Find an equation for f(x), the inverse function. The function f(x) = x*2, is one-to-one. f¹(x) = ₁x #6 (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Use the linear fit of the data to make the required predictions. 4. The boiling point of water is lower at higher elevations because of the lower atmospheric pressure. The boiling point of water in some different cities is given in the table and plotted in the graph along with a line of fit. Use the linear model to predict the boiling point of water in Santa Fe, New Mexico and Osaka, Japan. The altitude of Santa Fe, New Mexico is 7199 feet and the altitude of Osaka, Japan is 12 feet. City Altitude (feet) Boiling Point (°F) Chicago 597 210 Denver 5300 201 Kathmandu4600 205 Madrid 2188 207 Miami 6 210 201 The equation of the line of fit is y=-0.00167x + 211. Step 1- Evaluate the equation of the line of fit at the desired input values. The altitude inputs are x = ... and x =...

For f(x) = 10/x and g(x) =10/x , find the following functions. a. (fog)(x); b. (gof)(x); c. (fog)(6); d. (gof)(6) a. (fog)(x) =

A manufacturer has total revenue given by the function R(x) = 5.80x and has total cost given by C(x) = 1.80x + 76 where x is the number of units produced and sold. How many units must be produced to break-even?

Let the functions f and g be given by the equations on the right. Evaluate the indicated function without finding an equation for the function. f(x) = 3x - 1 g(x) = 5x-4 (fog)(1)=