0. Functions

State the inputs and outputs of the following relation. Is it a function? {$\left(-3,5\right),\left(0,2\right),\left(3,5\right)$}

State the inputs and outputs of the following relation. Is it a function? {$\left(2,5\right),\left(0,2\right),\left(2,9\right)$}

Find the domain and range of the following graph (write your answer using interval notation).

Test

Decide whether $f$, $g$, or both represent one-to-one functions.​​ <IMAGE>

Daylight function for 40 °N Verify that the function​​ $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.

Daylight function for 40 °N Verify that the function​​ $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at $t=172$  and $t=355$, respectively (corresponding to the solstices).

Daylight function for 40 °N Verify that the function​​ $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

$D\left(81\right)=12$ and $D\left(264\right)\approx 12$  (corresponding to the equinoxes).

The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is $s\left(t\right)=-16t^2+32t+48$ .

d. When does the stone strike the ground?

Demand and elasticity Based on sales data o​​ver the past year, the owner of a DVD store devises the demand function $D\left(p\right)=40-2p$ , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of $10?

Velocity fro​​​​m po​​sition ​The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

a. Assume the velocity of the object is 0 when $t=0$ . For what other values of t is the velocity of the object zero?

Velocity fro​​​​m po​​sition ​The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

c. Sketch a graph of the velocity function.

Fish length Assume the length L​​ (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

Evaluating functions from graphs Assume ƒ is an odd function and that both ƒ and g are one-to-one. Use the (incomplete) graph of ƒ and g the graph of to find the following function values. <IMAGE>

ƒ(g(4))

Slope functions Determine the slope function S (x) for the following functions

$\text{ƒ}\left(x\right)=\mid x\mid$

Let ƒ(x) = 1/ (x³+1).

Compute ƒ(2) and ƒ(y²).

The National Weather Service releases approximately $70,000$ radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about $1000$ ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point $6$ ft above the ground and that $5$ seconds later, it is $83$ ft above the ground. Let $f\left(t\right)$ represent the height (in feet) that the radiosonde is above the ground $t$ seconds after it is released. Evaluate $\frac{f\left(5\right)-f\left(0\right)}{5-0}$ and interpret the meaning of this quotient.

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Find the slope of the secant line that passes through points $A$ and $B$. Interpret your answer as an average rate of change over the interval $1\leq{t}\leq{3}$.

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Repeat the procedure outlined in part (a) for the secant line that passes through points $P$ and $Q$.

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Notice that the curve in the figure is horizontal for an interval of time near $t=5.5$ hr. Give a plausible explanation for the horizontal line segment.

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points $P$ and $Q$ on the graph of the function.

a. Sketch a graph of the function and the secant line through $P$ and $Q$.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After $t$ seconds, an object dropped from rest falls a distance $d=16t^2$, where $d$ is measured in feet and $2\leq{t}\leq{5}$.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

a. What is the domain of f (in terms of a)?

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

Functions and Graphs

Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = |x| - 2

In Exercises 19–32, find the (a) domain and (b) range.

____

𝔂 = -2 + √1 - x

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = 2e⁻ˣ - 3

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = tan(2x - π)

In Exercises 19–32, find the (a) domain and (b) range.

_____

𝔂 = -1 + ∛ 2 - x

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = 3 cos x + 4 sin x (Hint: A trig identity is required.)