?In a normal respiratory cycle the volume of air that moves into and out of the lungs is

Suppose the graph of \(f\) is given. Write equations for the graphs that are obtained from the graph of \(f\) as follows. (a) Shift \(3\) units upward. (b) Shift \(3\) units downward. (c) Shift \(3\) units to the right. (d) Shift \(3\) units to the left. (e) Reflect about the \(x\)-axis. (f ) Reflect about the \(y\)-axis. (g) Stretch vertically by a factor of \(3\). (h) Shrink vertically by a factor of \(3\). Equation Transcription: Text Transcription: f f 3 3 3 3 x y 3 3

Explain how each graph is obtained from the graph of \(y=f(x)\). (a) \(y=f(x)+8\) (b) \(y=f(????+8)\) (c) \(y=8f(????)\) (d) \(y=f(8????)\) (e) \(y=-f(????)-1\) (f) \(y=8 f\left(\frac{1}{8} x\right)\) Equation Transcription: Text Transcription: ???? ? ????(????) ???? ? ????(????) + 8 ???? ? ????(???? +8) ???? ? 8????(????) ???? ? ????(8????) ???? ? -????(????) - 1 ???? ? 8????(1/8x)

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) \(y=\frac{1}{3} f(x)\) (d) \(y=-f(x+4)\) (e) \(y=2 f(x+6)\) Equation Transcription: Text Transcription: y=f(x) y=f(x-4) y=f(x)+3 y=1/3f(x) f=-f(x+4) y=2f(x+6)

The graph of f is given. Draw the graphs of the following functions. (a) y = f(x) - 3 (b) y = f(x + 1) (c) \(y=\frac{1}{2} f(x)\) (d) y = -f(x)

The graph of \(f\) is given. Use it to graph the following functions. (a) \(y=f(2 x)\) (b) \(y=f\left(\frac{1}{2} x\right)\) (c) \(y=f(-x)\) (d) \(y=-f(-x)\) Equation Transcription: Text Transcription: f y=f(2x) y=f(12x) y=f(-x) y=-f(-x)

The graph of \(y=\sqrt{3 x-x^{2}}\) is given. Use transformations to create a function whose graph is as shown.

The graph of \(y=\sqrt{3 x-x^{2}}\) is given. Use transformations to create a function whose graph is as shown. \(y=\sqrt{3 x-x^{2}}\) Equation Transcription: Text Transcription: y = sqrt 3x-x^2

(a) How is the graph of \(y=1+\sqrt{x}\) related to the graph of \(y=\sqrt{x}\)? Use your answer and Figure 4(a) to sketch the graph of \(y=1+\sqrt{x}\). (b) How is the graph of \(y=5\ \sin\ \pi x\) related to the graph of \(y=\sin\ x\)? Use your answer and Figure 6 to sketch the graph of \(y=5\ \sin\ \pi x\). Equation Transcription: Text Transcription: y = 1 + sqrt x y = sqrt x y = 1 + sqrt x y = 5 sin pi x y = sin x y = 5 sin pi x

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=1+x^{2}\) Equation Transcription: Text Transcription: y =1 + x^2

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=(x+1)^{2}\) Equation Transcription: Text Transcription: y = (x + 1)^2

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=|x+2|\) Equation Transcription: Text Transcription: y = |x+2|

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=1-x^{3}\) Equation Transcription: Text Transcription: y = 1 - x^3

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=\frac{1}{x}+2\)

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=-\sqrt{x}-1\) Equation Transcription: Text Transcription: y = - sqrt x - 1

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=\sin\ 4x\) Equation Transcription: Text Transcription: y = sin 4x

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=1+\frac{1}{x^{2}}\) Equation Transcription: Text Transcription: y = 1 + 1/x^2

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=2+\sqrt{x+1}\) Equation Transcription: Text Transcription: y = 2 + sqrt x+1

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=-(x-1)^{2}+3\) Equation Transcription: Text Transcription: y = - (x - 1)^2 + 3

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=x^{2}-2 x+5\) Equation Transcription: Text Transcription: y = x^2 - 2x + 5

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=(x+1)^{3}+2\) Equation Transcription: Text Transcription: y = (x+1)^3 + 2

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=2-|x|\) Equation Transcription: Text Transcription: y = 2 - |x|

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=2-2\ \cos\ x\) Equation Transcription: Text Transcription: y = 2 - 2 cos x

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=3 \sin \frac{1}{2} x+1\)

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=\frac{1}{4}\ \tan\left(x-\frac{\pi}{4}\right)\) Equation Transcription: Text Transcription: y = 1/4 tan (x - pi/4)

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=|\cos\ \pi x|\) Equation Transcription: Text Transcription: y = |cos pi x|

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Table 1.2.3, and then applying the appropriate transformations. \(y=|\sqrt{x}-1|\) Equation Transcription: Text Transcription: y = |sqrt x - 1|

The city of New Orleans is located at latitude \(30^{\circ} \mathrm{N}\). Use Figure 9 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 the sun rises at 5:51 am and sets at 6:18 pm in New Orleans.

A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is \(4.0\), and its brightness varies by \(\pm 0.35\) magnitude. Find a function that models the brightness of Delta Cephei as a function of time. Equation Transcription: Text Transcription: 4.0 pm 0.35

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about \(2.0 \ m\) and at high tide it is about \(12.0 \ m\). The natural period of oscillation is about 12 hours and on a particular day, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth \(D(t)\) (in meters) as a function of time \(t\) (in hours after midnight) on that day. Equation Transcription: Text Transcription: 2.0 m 12.0 m D(t) t

In a normal respiratory cycle the volume of air that moves into and out of the lungs is about \(500 \ mL\). The reserve and residue volumes of air that remain in the lungs occupy about \(2000 \ mL\) 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. Equation Transcription: Text Transcription: 500 mL 2000 mL V(t)

(a) How is the graph of \(y=f(|x|)\) related to the graph of \(f\)? (b) Sketch the graph of \(y=\sin\ |x|\). (c) Sketch the graph of \(y=\sqrt{|x|}\). Equation Transcription: Text Transcription: y = f(|x|) f y = sin |x| y = sqrt |x|

Use the given graph of \(f\) to sketch the graph of \(y=1 / f(x)\). Which features of \(f\) are the most important in sketching \(y=1 / f(x)\)? Explain how they are used. Equation Transcription: Text Transcription: f y=1/f(x) f y=1/f(x)

Find (a) \(f+g\), (b) \(f-g\), (c) \(fg\), and (d) \(f/g\) and state their domains. \(f(x)=\sqrt{25-x^2},\quad\ \ \ g(x)=\sqrt{x+1}\) Equation Transcription: Text Transcription: f+g f-g fg f/g f(x) = sqrt 25-x^2, g(x) = sqrt x+1

Find (a) \(f+g\), (b) \(f-g\), (c) \(fg\), and (d) \(f/g\) and state their domains. \(f(x)=\frac{1}{x-1},\quad\ \ \ g(x)=\frac{1}{x}-2\) Equation Transcription: Text Transcription: f+g f-g fg f/g f(x) = 1/x-1, g(x) = 1/x -2

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=x^3+5,\quad\ \ \ g(x)=\sqrt[3]{x}\) Equation Transcription: Text Transcription: f circ g g circ f f circ f g circ g f(x) = x^3 + 5, g(x) = cube root x

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=\frac{1}{x},\quad\ \ \ g(x)=2x+1\) Equation Transcription: Text Transcription: f circ g g circ f f circ f g circ g f(x) = 1/x, g(x) = 2x + 1

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=\frac{1}{\sqrt{x}},\quad\ \ g(x)=x+1\)

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=\frac{x}{x+1},\quad\ \ \ g(x)=2x-1\) Equation Transcription: Text Transcription: f circ g g circ f f circ f g circ g f(x) = x/x+1, g(x) = 2x - 1

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=\frac{2}{x},\quad\ \ \ g(x)=\sin\ x\) Equation Transcription: Text Transcription: f circ g g circ f f circ f g circ g f(x) = 2/x, g(x) = sin x

Find the functions (a) \(f \circ g\), (b) \(g \circ f\) , (c) \(f \circ f\) , and (d) \(g \circ g\) and their domains. \(f(x)=\sqrt{5-x},\quad\ \ \ g(x)=\sqrt{x-1}\) Equation Transcription: Text Transcription: f circ g g circ f f circ f g circ g f(x) = sqrt 5-x, g(x) = sqrt x-1

Find \(f\ \circ\ g\ \circ\ h\). \(f(x)=3x-2,\quad\ \ \ g(x)=\sin\ x,\quad\ \ \ h(x)=x^2\) Equation Transcription: Text Transcription: f circ g circ h f(x) = 3x-2, g(x) = sin x, h(x) = x^2

Find \(f\ \circ\ g\ \circ\ h\). \(f(x)=|x-4|,\quad\ \ \ g(x)=2^x,\quad\ \ \ h(x)=\sqrt{x}\) Equation Transcription: Text Transcription: f circ g circ h f(x) = |x-4|, g(x) = 2^x, h(x) = sqrt x

Find \(f\ \circ\ g\ \circ\ h\). \(f(x)=\sqrt{x-3},\quad\ \ \ g(x)=x^2,\quad\ \ \ h(x)=x^3+2\) Equation Transcription: Text Transcription: f circ g circ h f(x) = sqrt x-3, g(x) = x^2, h(x) = x^3 + 2

Find \(f\ \circ\ g\ \circ\ h\). \(f(x)=\tan\ x,\quad\ \ \ g(x)=\frac{x}{x-1},\quad\ \ \ h(x)=\sqrt[3]{x}\) Equation Transcription: Text Transcription: f circ g circ h f(x) = tan x, g(x) = x/x-1, h(x) = cuberoot x

Express the function in the form \(f\ \circ\ g\). \(F(x)=\left(2 x+x^{2}\right)^{4}\)

Express the function in the form \(f\ \circ\ g\). \(F(x)=\cos ^{2} x\) Equation Transcription: Text Transcription: f circ g F(x) = cos^2 x

Express the function in the form \(f\ \circ\ g\). \(F(x)=\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}\) Equation Transcription: Text Transcription: f circ g F(x) = cuberoot x / 1 + cuberoot x

Express the function in the form \(f \circ g\). \(G(x)=\sqrt[3]{\frac{x}{1+x}}\)

Express the function in the form \(f\ \circ\ g\). \(v(t)=\sec\left(t^2\right)\ \tan\left(t^2\right)\) Equation Transcription: Text Transcription: f circ g v(t) = sec(t^2) tan(t^2)

Express the function in the form \(f\ \circ\ g\). \(H(x)=\sqrt{1+\sqrt{x}}\) Equation Transcription: Text Transcription: f circ g H(x) = sqrt 1 + sqrt x

Express the function in the form \(f\ \circ\ g\ \circ\ h\). \(R(x)=\sqrt{\sqrt{x}-1}\) Equation Transcription: Text Transcription: f circ g circ h R(x) = sqrt sqrt x - 1

Express the function in the form \(f\ \circ\ g\ \circ\ h\). \(H(x)=\sqrt[8]{2+|x|}\) Equation Transcription: Text Transcription: f circ g circ h H(x) = 8th root 2+|x|

Express the function in the form \(f\ \circ\ g\ \circ\ h\). \(S(t)=\sin^2(\cos\ t)\) Equation Transcription: Text Transcription: f circ g circ h S(t) = sin^2 (cos t)

Express the function in the form \(f\ \circ\ g\ \circ\ h\). \(H(t)=\cos(\sqrt{\tan\ t}+1)\) Equation Transcription: Text Transcription: f circ g circ h H(t) = cos(sqrt tan t + 1)

Use the table to evaluate each expression. (a) f(g(3)) (b) g(f(2)) (c) \((f\ \circ\ g)(5)\) (d) \((g\ \circ\ f)(5)\)

Use the table to evaluate each expression. (a) \(g(g(g(2)))\) (b) \((f\ \circ\ f\ \circ\ f)(1)\) (c) \((f\ \circ\ f\ \circ\ g)(1)\) (d) \((g\ \circ\ f\ \circ\ g)(3)\) Equation Transcription: Text Transcription: g(g(g(2))) (f circ f circ f)(1) (f circ f circ g)(1) (g circ f circ g)(3)

Use the given graphs of \(f\) and \(g\) to evaluate each expression, or explain why it is undefined. (a) \(f(g(2))\) (b) \(g(f(0))\) (c) \((f\ \circ\ g)(0)\) (d) \((g\ \circ\ f)(6)\) (e) \((g\ \circ\ g)(-2)\) (f) \((f\ \circ\ f)(4)\) Equation Transcription: Text Transcription: f g f(g(2)) g(f(0)) (f circ g)(0) (g circ f)(6) (g circ g)(-2) (f circ f)(4)

Use the given graphs of \(f\) and \(g\) to estimate the value of \(f(g(x))\) for \(x=-5,\ -4,\ -3,\ldots,\ 5\). Use these estimates to sketch a rough graph of \(f\ \circ\ g\). Equation Transcription: Text Transcription: f g f(g(x)) x = -5, -4, -3 . . . , 5 f circ g

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of \(60 \ cm/s\). (a) Express the radius \(r\) of this circle as a function of the time \(t\) (in seconds). (b) If \(A\) is the area of this circle as a function of the radius, find \(A\ \circ \ r\) and interpret it. Equation Transcription: Text Transcription: 60 cm/s r t A A circ r

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of \(2 \ cm/s\). (a) Express the radius \(r\) of the balloon as a function of the time \(t\) (in seconds). (b) If \(V\) is the volume of the balloon as a function of the radius, find \(V\ \circ\ r\) and interpret it. Equation Transcription: Text Transcription: 2 cm/s r t V V circ r

A ship is moving at a speed of \(30 \ km/h\) parallel to a straight shoreline. The ship is \(6 \ km\) from shore and it passes a lighthouse at noon. (a) Express the distance \(s\) between the lighthouse and the ship as a function of \(d\), the distance the ship has traveled since noon; that is, find \(f\) so that \(s=f(d)\). (b) Express \(d\) as a function of \(t\), the time elapsed since noon; that is, find \(g\) so that \(d=g(t)\). (c) Find \(f\ \circ\ g\). What does this function represent? Equation Transcription: Text Transcription: 30 km/h 6 km s d f s = f(d) d t g d = g(t) f circ g

An airplane is flying at a speed of \(350 \ mi/h\) at an altitude of one mile and passes directly over a radar station at time \(t=0\). (a) Express the horizontal distance \(d\) (in miles) that the plane has flown as a function of \(t\). (b) Express the distance \(s\) between the plane and the radar station as a function of \(d\). (c) Use composition to express \(s\) as a function of \(t\). Equation Transcription: Text Transcription: 350 mi/h t = 0 d t s d s t

The Heaviside function H is defined by \(H(t)=\left\{\begin{array}{ll} 0 & \ \ \ \text { if } t<0 \\ 1 & \ \ \ \text { if } t \geqslant 0 \end{array}\right.\) It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 5 corresponds to a translation.)

The Ramp Function The Heaviside function defined in Exercise 63 can also be used to define the ramp function \(y=ctH(t)\), which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function \(y=tH(t)\). (b) Sketch the graph of the voltage \(V(t)\) in a circuit if the switch is turned on at time \(t=0\) and the voltage is gradually increased to \(120\) volts over a \(60\)-second time interval. Write a formula for \(V(t)\) in terms of \(H(t)\) for \(t \leqslant 60\). (c) Sketch the graph of the voltage \(V(t)\) in a circuit if the switch is turned on at time \(t=7\) seconds and the voltage is gradually increased to \(100\) volts over a period of \(25\) seconds. Write a formula for \(V(t)\) in terms of \(H(t)\) for \(t \leqslant 32\). Equation Transcription: ? ? Text Transcription: y=ctH(t) y=tH(t) V(t) t=0 120 60 V(t) H(t) t leq 60 V(t) t=7 100 25 V(t) H(t) t leq 32

Let \(f\) and \(g\) be linear functions with equations \(f(x)=m_{1} x+b_{1}\) and \(g(x)=m_{2} x+b_{2}\). Is \(f\ \circ\ g\) also a linear function? If so, what is the slope of its graph? Equation Transcription: Text Transcription: f g f(x) = m_1 x + b_1 g(x) = m_2 x + b_2 f circ g

If you invest x dollars at \(4 \%\) interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.04 x. Find \(A\ \circ\ A\), \(A\ \circ\ A\ \circ\ A\), and \(A\ \circ\ A\ \circ\ A\ \circ\ A\). What do these compositions represent? Find a formula for the composition of n copies of A.

(a) If \(g(x)=2 x+1\) and \(h(x)=4 x^{2}+4 x+7\), find a function \(f\) such that \(f\ \circ\ g=h\). (Think about what operations you would have to perform on the formula for \(g\) to end up with the formula for \(h\).) (b) If \(f(x)=3 x+5\) and \(h(x)=3 x^{2}+3 x+2\), find a function \(g\) such that \(f\ \circ\ g=h\). Equation Transcription: Text Transcription: g(x) = 2x + 1 h(x) = 4x^2 + 4x + 7 f f circ g = h g h f(x) = 3x + 5 h(x) = 3x^2 + 3x + 2 g f circ g = h

If \(f(x)=x+4\) and \(h(x)=4 x-1\), find a function \(g\) such that \(g\ \circ\ f=h\). Equation Transcription: Text Transcription: f(x) = x + 4 h(x) = 4x - 1 g g circ f=h

Suppose \(g\) is an even function and let \(h=f\ \circ\ g\). Is \(h\) always an even function? Equation Transcription: Text Transcription: g h = f circ g h

Suppose \(g\) is an odd function and let \(h=f\ \circ\ g\). Is \(h\) always an odd function? What if \(f\) is odd? What if \(f\) is even? Equation Transcription: Text Transcription: g h = f circ g h f f

Let \(f(x)\) be a function with domain \(\mathbb{R}\). (a) Show that \(E(x)=f(x)+f(-x)\) is an even function. (b) Show that \(O(x)=f(x)-f(-x)\) is an odd function. (c) Prove that every function \(f(x)\) can be written as a sum of an even function and an odd function. (d) Express the function \(f(x)=2^{x}+(x-3)^{2}\) as a sum of an even function and an odd function. Equation Transcription: Text Transcription: f(x) R E(x) = f(x) + f(-x) O(x) = f(x) - f(-x) f(x) f(x) = 2^x + (x - 3)^2