Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. y = f(x−2) y = 2f(x) y = f(x)+2 y = 1 2 − f(x) y = f(x)−2 (b) Shift 2 units downward. y = 1 2 − f(x) y = f(x−2) y = f(x)+2 y = 2 f(x) y = f(x)−2 (c) Shift 2 units to the right. y = f(x−2) y = 2 f(x) y = 1−f(x) y = f(x)+2 y = f(x)−2 (d) Shift 2 units to the left. y = f(x−2) y = f(x+2) y = f(x)−2 y = 1 2 − f(x) y = f(x)+2 (e) Reflect about the x-axis. y = f(x)+2 y = f(x−2) y = 1 2 −f(x) y = −f(x) y = 2f(x) (f) Reflect about the y-axis. y = f(x−2) y = 1 2 −f(x) y = f(x)+2 y = 2f(x) y = f(−x) (g) Stretch vertically by a factor of 2 . y = 1 2 −f(x) y = 2f(x) y = f(x)+2 y = f(x−2) y = f(x)−2 (h) Shrink vertically by a factor of 2 . y = f(x)−2 y = f(x)+2 y = f(x−2) y = 2 f(x) y = 12−f(x)
Explain how each graph is obtained from the graph of y = f(x). (a) y = f(x)+6 Stretch the graph horizontally and vertically by a factor of 6 . Shift the graph 6 units to the right. Shift the graph 6 units upward. Shift the graph 6 units downward. Shift the graph 6 units to the left. (b) y = f(x+6) Shift the graph 6 units to the right. Shift the graph 6 units upward. Stretch the graph horizontally and vertically by a factor of 6 . Shift the graph 6 units downward. Shift the graph 6 units to the left. (c) y = 6f(x) Shift the graph 6 units upward. Shrink the graph horizontally by a factor of 6 . Stretch the graph horizontally and vertically by a factor of 6 . Stretch the graph vertically by a factor of 6 . Shift the graph 6 units to the left. (d) y = f(6x) Shift the graph 6 units upward. Shift the graph 6 units to the left. Stretch the graph vertically by a factor of 6 . Stretch the graph horizontally and vertically by a factor of 6 . Shrink the graph horizontally by a factor of 6 . (e) y = −f(x)−1 First reflect the graph about the y-axis, and then shift it 1 unit downward. First reflect the graph about the y-axis, and then shift it 1 unit upward. First reflect the graph about the x-axis, and then shift it 1 unit downward. First reflect the graph about the x-axis, and then shift it 1 unit upward. First reflect the graph about the x-axis, and then shift it 1 unit left. (f) y = 6f(16 x) Shrink the graph horizontally by a factor of 6. Stretch the graph horizontally and vertically by a factor of 6 . Stretch the graph vertically by a factor of 6 . Shrink the graph horizontally and vertically by a factor of 6. Stretch the graph horizontally by a factor of 6.
The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices. (a) y = f(x − 4) (b) y = f(x) + 3 (c) y = 1 3 f(x) (d) y = −f(x + 4) (e) y = 2f(x + 6)
Find the domain of each function. (Enter your answer using interval notation.) (a) f(x) = 25−ex2 1−e25−x2 (b) f(x) = 1+x ecosx
The graph of y = 4x − x2 is given. Use transformations to create a function whose graph is as shown. y =
The city of New Orleans is located at latitude 30∘N. Use the figure below to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31(t = 90) the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans. L(t) = Graph of the length of daylight from March 21 through December 21 at various latitudes
Use the table to evaluate each expression. (a) f(g(1)) (b) g(f(1)) (c) f(f(1)) (d) g(g(1)) (e) (g∘f)(3) (f) (f∘g)(6)
Find the domain of each function. (Enter your answers using interval notation.) (a) f(x) = 9 6+ex (b) f(x) = 3 1−ex
Find each of the following functions. f(x) = 2−2x, g(x) = cos(x) (a) f∘g State the domain of the function. (Enter your answer using interval notation.) (b) g∘f State the domain of the function. (Enter your answer using interval notation.) (c) f∘f State the domain of the function. (Enter your answer using interval notation.) (d) g∘g State the domain of the function. (Enter your answer using interval notation.)
Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x) = x3 + 5x2, g(x) = 6x2 − 1 (a) f+g f+g = domain (b) f−g f−g = domain (c) fg fg = domain (d) f/g f/g = domain
Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x) = 8x + 3, g(x) = x2 + x (a) f∘g (f∘g)(x) = domain (b) g∘f (g∘f)(x) = domain (c) f∘f (f∘f)(x) = domain (d) g∘g (g∘g)(x) = domain
The graph of the function y = 5x − x2 is given. Use transformations to create a function whose graph is as shown. y =
Find each of the following functions. f(x) = x2 − 3, g(x) = 4x + 4 (a) f∘g State the domain of the function. (Enter your answer using interval notation.) (b) g∘f State the domain of the function. (Enter your answer using interval notation.) (c) f∘f State the domain of the function. (Enter your answer using interval notation.) (d) g∘g State the domain of the function. (Enter your answer using interval notation.)
Find each of the following functions. f(x) = x 1+x, g(x) = sin(5x) (a) f∘g State the domain of the function. x ≠ 3π7 + 25πn [where n is any integer] x ≠ 3π20 + 25πn [where n is any integer] x ≠ 3π10 + 25πn [where n is any integer] x ≠ 3π10 + 7πn [where n is any integer] x ≠ 7π10 + 7πn [where n is any integer] (b) g∘f State the domain of the function. (Enter your answer using interval notation.) (c) f∘f State the domain of the function. (Enter your answer using interval notation.) (d) g∘g State the domain of the function. (Enter your answer using interval notation.)
Consider the following functions. f(x) = x x+1, g(x) = 1 x Find (f∘g)(x) Find the domain of (f∘g)(x). (Enter your answer using interval notation.) Find (g∘f)(x). Find the domain of (g∘f)(x). (Enter your answer using interval notation.) Find (f∘f)(x) Find the domain of ( f∘f)(x). (Enter your answer using interval notation.) Find (g∘g)(x) Find the domain of (g∘g)(x). (Enter your answer using interval notation.)
Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x) = x3 + 3x2, g(x) = 7x2 − 3 (a) f+g f+g = domain (b) f−g f−g = domain (c) fg fg = domain (d) f/g f/g = domain
Find each of the following functions. f(x) = 5 − x, g(x) = x2 − 9 (a) f+g State the domain of the function. (Enter your answer using interval notation.) (b) f−g State the domain of the function. (Enter your answer using interval notation.) (c) fg State the domain of the function. (Enter your answer using interval notation.) (d) f/g State the domain of the function. (Enter your answer using interval notation.)
A variable star is one whose brightness alternately increases and decreases. For one such star, the time between periods of maximum brightness is 4.5 days, the average brightness (or magnitude) of the star is 5.1 , and its brightness varies by ±0.30 magnitude. Find a function that models the brightness of the star as a function of time (in days), t. (Assume that at t = 0 the brightness of the star is 5.1 and that it is increasing.) f(t) =
In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 504 mL. The reserve and residual volumes of air that remain in the lungs occupy about 1955 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air V(t) in the lungs as a function of time. V(t) =
Use the given graphs of f and g to evaluate each expression, or if the expression is undefined, enter UNDEFINED. (a) f(g(2)) (b) g(f(0)) (c) (f∘g)(0) (d) (g∘f)(6) (e) (g∘g)(−2) (f) (f∘f)(4)
Suppose the graphs of f(x) = x2 and g(x) = 2 x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that, at a distance 2 ft to the right of the origin, the height of the graph of f is 48 ft but the height of the graph of g is about 265 mi. (Round your final answer to the nearest whole number.) First, 2 ft = in. Therefore, f(24) = in = ft. However, g(24) 2 12 5280 mi