함수와 그래프 전체 문제

A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

Near the surface of the moon, the distance that an object falls is a function of time. It is given by $d(t)=2.6667{t}^{2},$ where $t$ is in seconds and $d(t)$ is in feet. If an object is dropped from a certain height, find the average velocity of the object from $t=1$ to $t=2.$

The graph in (the referenced figure) illustrates the decay of a radioactive substance over $t$ days.

Use the graph to estimate the average decay rate from $t=5$ to $t=15.$

How does one find the domain of the quotient of two functions, $\frac{f}{g}?$

What is the composition of two functions, $f\circ g?$

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

How do you find the domain for the composition of two functions, $f\circ g?$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)={x}^{2}+2x$ and $g(x)=6-{x}^{2},$ find $f+g,\,f-g,\,fg,$ and $\,\frac{f}{g}.$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)=-3{x}^{2}+x$ and $g(x)=5,$ find $f+g,\,f-g,\,fg,$ and $\,\frac{f}{g}.$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)=2{x}^{2}+4x$ and $g(x)=\frac{1}{2x},$ find $f+g,\,f-g,\,fg,$ and $\frac{f}{g}.$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)=\frac{1}{x-4}$ and $g(x)=\frac{1}{6-x},$ find $f+g,\,f-g,\,fg,$ and $\frac{f}{g}.$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)=3{x}^{2}$ and $g(x)=\sqrt{x-5},$ find $f+g,\,f-g,\,fg,$ and $\frac{f}{g}.$

For the following exercises, determine the domain for each function in interval notation.

Given $f(x)=\sqrt{x}$ and $g(x)=|x-3|,$ find $\frac{g}{f}.$

For the following exercises, determine the domain for each function in interval notation.

For the following exercise, find the indicated function given $f(x)=2{x}^{2}+1$ and $g(x)=3x-5.$

• ⓐ $f(g(2))$
• ⓑ $f(g(x))$
• ⓒ $g(f(x))$
• ⓓ $(g\circ g)(x)$
• ⓔ $(f\circ f)(-2)$

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

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

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

$f(x)=\sqrt{x}+2,\,g(x)={x}^{2}+3$

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

$f(x)=|x|,\,g(x)=5x+1$

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

$f(x)=\sqrt[3]{x},\,g(x)=\frac{x+1}{{x}^{3}}$

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

$f(x)=\frac{1}{x-6},\,g(x)=\frac{7}{x}+6$

For the following exercises, use each pair of functions to find $f(g(x))$ and $g(f(x)).$ Simplify your answers.

$f(x)=\frac{1}{x-4},\,g(x)=\frac{2}{x}+4$