합성함수와 역함수

For the following exercises, use composition to determine which pairs of functions are inverses.

$f(x)=\frac{1}{x-1},x\ne 1,g(x)=\frac{1}{x}+1,x\ne 0$

For the following exercises, use composition to determine which pairs of functions are inverses.

$f(x)={x}^{3}+1,g(x)={(x-1)}^{1/3}$

For the following exercises, use composition to determine which pairs of functions are inverses.

$f(x)={x}^{2}+2x+1,x\ge -1,\quad g(x)=-1+\sqrt{x},x\ge 0$

For the following exercises, use composition to determine which pairs of functions are inverses.

$f(x)=\sqrt{4-{x}^{2}},0\le x\le 2,g(x)=\sqrt{4-{x}^{2}},0\le x\le 2$

For the following exercises, evaluate the functions. Give the exact value.

${\tan }^{-1}\left(\frac{\sqrt{3}}{3}\right)$

For the following exercises, evaluate the functions. Give the exact value.

${\cos }^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

For the following exercises, evaluate the functions. Give the exact value.

${\cot }^{-1}(1)$

For the following exercises, evaluate the functions. Give the exact value.

${\sin }^{-1}(-1)$

For the following exercises, evaluate the functions. Give the exact value.

${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

For the following exercises, evaluate the functions. Give the exact value.

$\cos \left({\tan }^{-1}\left(\sqrt{3}\right)\right)$

For the following exercises, evaluate the functions. Give the exact value.

$\sin \left({\cos }^{-1}\left(\frac{\sqrt{2}}{2}\right)\right)$

For the following exercises, evaluate the functions. Give the exact value.

${\sin }^{-1}\left(\sin \left(\frac{\pi }{3}\right)\right)$

For the following exercises, evaluate the functions. Give the exact value.

${\tan }^{-1}\left(\tan \left(-\frac{\pi }{6}\right)\right)$

The function $C=T(F)=(5/9)(F-32)$ converts degrees Fahrenheit to degrees Celsius.

a. Find the inverse function $F={T}^{-1}(C)$ b. What is the inverse function used for?

[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function $V=f(x)=500(0.04-{x}^{2})$ for $0\le x\le 0.2.$

a. Find $x={f}^{-1}(V).$ b. Interpret what the inverse function is used for. c. Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.

A function that converts dress sizes in the United States to those in Europe is given by $D(x)=2x+24.$

a. Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States. b. Find the function that converts European dress sizes to U.S. dress sizes. c. Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.

[T] The cost to remove a toxin from a lake is modeled by the function

$C(p)=75p/(85-p),$ where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.

a. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake. b. Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.

[T] A race car is accelerating at a velocity given by

$v(t)=\frac{25}{4}t+54,$ where v is the velocity (in feet per second) at time t.

a. Find the velocity of the car at 10 sec. b. Find the inverse function. c. Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.

[T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\mu =2{\sin }^{-1}\left(\frac{1}{M}\right).$

Find the Mach angle (to the nearest degree) for the following Mach numbers.

a. $M=1.4$ b. $M=2.8$ c. $M=4.3$

[T] Using $\mu =2{\sin }^{-1}\left(\frac{1}{M}\right),$ find the Mach number M for the following angles.

a. $\mu =\frac{\pi }{6}$ b. $\mu =\frac{2\pi }{7}$ c. $\mu =\frac{3\pi }{8}$