0. Functions

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.

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

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Let $r\left(t\right)$ be the vector-valued function defined by $r\left(t\right)=(3t,\frac{1}{t+1},e^{-t})$. Evaluate the definite integral from $t=0$ to $t=1$ of $r\left(t\right)dt$.

Finding Formulas for Functions

A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.

Let ƒ(x) = 1/ (x³+1).

Compute ƒ(2) and ƒ(y²).

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 $Q$.

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

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𝔂 = -1 + ∛ 2 - x

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

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

𝔂 = 2 + 3x² .

x² + 4

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.

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.

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.

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

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

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

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{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)?