함수와 그래프 전체 문제

For the following exercises, use a graph to help determine the domain of the functions.

$f(x)=\sqrt{\frac{{x}^{2}-x-20}{x-2}}$

For the following exercises, use a graph to help determine the domain of the functions.

$f(x)=\sqrt{\frac{9-{x}^{2}}{x+4}}$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f(x)={x}^{3}-x-2,\,y=1,\,2,\,3$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f(x)={x}^{3}+x-2,\,y=0,\,1,\,2$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f(x)={x}^{3}+3x-4,\,y=0,\,1,\,2$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f(x)={x}^{3}+8x-4,\,y=-1,\,0,\,1$

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f(x)={x}^{4}+5x+1,\,y=-1,\,0,\,1$

For the following exercises, find the inverse of the functions with $a,b,c$ positive real numbers.

$f(x)=a{x}^{3}+b$

For the following exercises, find the inverse of the functions with $a,b,c$ positive real numbers.

$f(x)={x}^{2}+bx$

For the following exercises, find the inverse of the functions with $a,b,c$ positive real numbers.

$f(x)=\sqrt{a{x}^{2}+b}$

For the following exercises, find the inverse of the functions with $a,b,c$ positive real numbers.

$f(x)=\sqrt[3]{ax+b}$

For the following exercises, find the inverse of the functions with $a,b,c$ positive real numbers.

$f(x)=\frac{ax+b}{x+c}$

For the following exercises, determine the function described and then use it to answer the question.

An object dropped from a height of 200 meters has a height, $h(t),$ in meters after $t$ seconds have lapsed, such that $h(t)=200-4.9{t}^{2}.$ Express $t$ as a function of height, $h,$ and find the time to reach a height of 50 meters.

For the following exercises, determine the function described and then use it to answer the question.

An object dropped from a height of 600 feet has a height, $h(t),$ in feet after $t$ seconds have elapsed, such that $h(t)=600-16{t}^{2}.$ Express $t$ as a function of height $h,$ and find the time to reach a height of 400 feet.

For the following exercises, determine the function described and then use it to answer the question.

The volume, $V,$ of a sphere in terms of its radius, $r,$ is given by $V(r)=\frac{4}{3}\pi {r}^{3}.$ Express $r$ as a function of $V,$ and find the radius of a sphere with volume of 200 cubic feet.

For the following exercises, determine the function described and then use it to answer the question.

The surface area, $A,$ of a sphere in terms of its radius, $r,$ is given by $A(r)=4\pi {r}^{2}.$ Express $r$ as a function of $A,$ and find the radius of a sphere with a surface area of 1000 square inches.

For the following exercises, determine the function described and then use it to answer the question.

A container holds 100 ml of a solution that is 25 ml acid. If $n$ ml of a solution that is 60% acid is added, the function $C(n)=\frac{25+.6n}{100+n}$ gives the concentration, $C,$ as a function of the number of ml added, $n.$ Express $n$ as a function of $C$ and determine the number of mL that need to be added to have a solution that is 50% acid.

For the following exercises, determine the function described and then use it to answer the question.

The period $T,$ in seconds, of a simple pendulum as a function of its length $l,$ in feet, is given by $T(l)=2\pi \sqrt{\frac{l}{32.2}}$ . Express $l$ as a function of $T$ and determine the length of a pendulum with period of 2 seconds.

For the following exercises, determine the function described and then use it to answer the question.

The volume of a cylinder , $V,$ in terms of radius, $r,$ and height, $h,$ is given by $V=\pi {r}^{2}h.$ If a cylinder has a height of 6 meters, express the radius as a function of $V$ and find the radius of a cylinder with volume of 300 cubic meters.

For the following exercises, determine the function described and then use it to answer the question.

The surface area, $A,$ of a cylinder in terms of its radius, $r,$ and height, $h,$ is given by $A=2\pi {r}^{2}+2\pi rh.$ If the height of the cylinder is 4 feet, express the radius as a function of $A$ and find the radius if the surface area is 200 square feet.