Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. (a) f(x) = log7(x) trigonometric function root function exponential function logarithmic function polynomial function of degree 2 rational function power function algebraic function (b) g(x) = x 4 logarithmic function trigonometric function rational function root function exponential function polynomial function of degree 2 (c) h(x) = 2x3 7−x2 trigonometric function polynomial of degree 2 power function logarithmic function exponential function rational function root function (d) u(t) = 1 − 1.1t + 2.51t2 exponential function trigonometric function root function logarithmic function power function polynomial function of degree 2 (e) v(t) = 4 t root function exponential function trigonometric function logarithmic function power function (f) w(θ) = sin(θ)cos5(θ) power function logarithmic function rational function exponential function algebraic function polynomial function of degree 2 root function trigonometric function
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in the figures below and, if necessary, the transformations of Section 1.3. y = −4 −x
(a) Write an equation that defines the exponential function with base b > 0. f(x) = bx f(x) = logb(x) f(x) = bex f(x) = eb/x f(x) = xb (b) What is the domain of this function? (Enter your answer using interval notation.) (c) If b ≠ 1, what is the range of this function? (Enter your answer using interval notation.) (d) Sketch the general shape of the graph of the exponential function for each of the following cases. (i) b > 1 (ii) b = 1 (iii) 0 < b < 1
Find the domain of the function. f(x) = 3 cos(x) 1−sin(x) {x | x ≠ π 2 + nπ, n an integer } {x | x ≠ π 2 + 2nπ, n an integer} {x | x ≠ π 4 + 2nπ, n an integer} {x ∣ x ≠ π + 2nπ, n an integer } {x | x ≠ π 4 + nπ, n an integer}
A graphing calculator is recommended. Graph the given functions on a common screen. How are these graphs related? y = 2 x, y = ex, y = 5 x, y = 20 x All of these graphs approach as x→−∞, all of them pass through the point (x, y) = ( ), and all of them are increasing and approach ∞ as x → ∞. The larger the base, the the function increases for x > 0, and the it approaches 0 as x → −∞.
The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = 9 5 C + 32. (a) Sketch a graph of this function. (b) What is the slope of the graph? What does it represent? The slope means that F increases degrees for each increase of 1∘C. What is the F-intercept? What does it represent? The F-intercept of is the Fahrenheit temperature corresponding to a Celsius temperature of 0.
Jason leaves Detroit at 9:00 PM and drives at a constant speed west along I-94. He passes Ann Arbor, 40 mi from Detroit, at 9:48 PM. (a) Express the distance d traveled in terms of the time t (in hours) elapsed. d(t) = (b) Draw the graph of the equation in part (a). (c) What is the slope of this line?
The manager of a furniture factory finds that it costs $2600 to manufacture 60 chairs in one day and $4800 to produce 260 chairs in one day. (a) Express the cost C (in dollars) as a function of the number of chairs x produced, assuming that it is linear. C = Sketch the graph. (b) What is the slope of the graph? What does it represent? It represents the cost (in dollars) of operating the factory daily. It represents the cost (in dollars) of producing each additional chair. It represents the time (in days) to produce each additional chair. It represents the number of chairs produced. (c) What is the y-intercept of the graph? What does it represent? It represents the fixed daily cost (in dollars) of operating the factory. It represents the time (in days) to produce each additional chair. It represents the number of chairs produced. It represents the cost (in dollars) of producing each additional chair.
At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface, the water pressure increases by 4.34 lb/in2 for every 10 ft of descent. (a) Express the water pressure P (in Ib/in2) as a function of the depth d in feet below the ocean surface. P(d) = Ib/in2 (b) At what depth (in ft ) is the pressure 50 lb/in2? (Round your answer to the nearest whole number.) ft
The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $444 to drive 380 mi and in June it cost her $600 to drive 900 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. C(d) = (b) Use part (a) to predict the cost of driving 1900 miles per month. $ (c) Draw the graph of the linear function. What does the slope represent? It represents the distance (in miles) traveled. It represents the cost (in dollars) of driving. It represents the fixed cost (amount she pays even if she does not drive). It represents the cost (in dollars) per mile. (d) What does the C-intercept represent? It represents the cost (in dollars) per mile. It represents the fixed cost (amount she pays even if she does not drive). It represents the distance (in miles) traveled. It represents the cost (in dollars) of driving. (e) Why does a linear function give a suitable model in this situation? A linear function is suitable because the monthly cost increases even if the miles driven is constant. A linear function is suitable because the monthly cost is fixed despite the fact that the miles driven may vary. A linear function is suitable because the monthly cost increases as the number of miles driven decreases. A linear function is suitable because the monthly cost increases as the number of miles driven increases.
Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70∘F and 153 chirps per minute at 80∘F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. T = (b) What is the slope of the graph? What does it represent? The slope means that the temperature in Fahrenheit degrees increases as rapidly as the number of cricket chirps per minute. (c) If the crickets are chirping at 200 chirps per minute, estimate the temperature. (Round your answer to the nearest whole number.) ∘F
(a) Find an equation for the family of linear functions with slope 2. (Use the standard coordinate variables x and y. You may use m for the slope and b for the y-intercept as needed.) Sketch several members of the family. (b) Find an equation for the family of linear functions such that f(2) = 1. (Use the standard coordinate variables x and y. You may use in for the slope and b for the y-intercept as needed.) Sketch several members of the family. (c) What equation belongs to both families? (Use the standard coordinate variables x and y.) Need Help?
Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator. ) (a) y = x2 f h g (b) y = x5 g f h (c) y = x8 h g f
Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator. ) (a) y = 5x f F g G (b) y = 5 x f G g F (c) y = x5 G f F g (d) y = x5 F G g f
Recent studies indicate that the average surface temperature of a planet has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.02t + 8.55, where T is temperature in ∘C and t represents years since 1900. (a) What do the slope and T-intercept represent? The slope is, which means that the average surface temperature of the planet is at a rate of ∘C per year. The T-intercept is, which presents the average surface temperature in the year. (b) Use the equation to predict the planet's average surface temperature in 2010. ∘C
If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c = 0.0417D(a + 1). Suppose the dosage for an adult is 50 mg. (a) Find the slope of the graph of c. (Round your answer to two decimal places.) What does it represent? The slope represents the of the dosage for a child for each change of 1 year in age. (b) What is the dosage for a newborn? (Round your answer to two decimal places.) mg
For each scatter plot, decide what type of function you might choose as a model for the data. (a) logarithmic function, of the form f(x) = aln(bx) + c exponential function, of the form f(x) = a ebx+c trigonometric function, of the form f(x) = a cos(bx)+c linear function, of the form f(x) = mx + b root function, of the form f(x) = a bx+c (b) linear function, of the form f(x) = mx + b root function, of the form f(x) = a bx + c trigonometric function, of the form f(x) = a cos(bx) + c exponential function, of the form f(x) = a eax + c logarithmic function, of the form f(x) = a ln(bx) − c
Many physical quantities are connected by inverse square laws, that is, by power functions of the form f(x) = kx−2. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light? times as bright
A graphing calculator is recommended. Graph the given functions on a common screen. How are these graphs related? y = 4 x, y = 10 x, y = (1 4)x, y = (1 10)x The functions with bases greater than 1(4 x and 10 x ) are , while those with bases less than 1[(1 4)x and (1 10)x] are. The graph of (1 4)x is the reflection of that of 4 x about the -axis, and the graph of (1 10)x is the reflection of that of 10 x about the -axis. The graph of 10 x increases more quickly than that of 4 x for x 0, and approaches 0 faster as ? V → −∞.