Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ o g

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$, which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$, where $t$ is the number of hours after the first observation.

How long does it take the population to reach $10,000$?

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$

Parabola properties Consider the general quadratic function ƒ(x) = ax² + bx + c , with a ≠ 0.

a. Find the coordinates of the vertex of the graph of the parabola y= ƒ(x)  in terms of a, b, and c.

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x\mid x\mid$

Prove the following identities.

$\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$

Prove the following identities.

$\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$

Composition of even and odd functions from graphs Assume ƒ is an even function and g is an odd function. Use the (incomplete) graphs of ƒ and g  in the figure to determine the following function values. <IMAGE>

a. ƒ(g(-2))

Composition of even and odd functions from graphs Assume ƒ is an even function and g is an odd function. Use the (incomplete) graphs of ƒ and g  in the figure to determine the following function values. <IMAGE>

e. g(g(-7))

Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

a. ƒ(g(-1))

Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

c. ƒ(g(-3))

Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

e. g(g(-1))

Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

g. ƒ (g(g(-2)))

Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

i. g(g(g(-1)))

Express $\theta$ in terms of $x$ using the inverse sine, inverse tangent, and inverse secant functions. <IMAGE>

{Use of Tech} Sum of squared integers Let T (n) = 1² + 2² + ... + n², where n is a positive integer. It can be shown that T (n) = n (n + 1) (2n + 1) / 8

c. What is the least value of n for which T(n) > 1000?

Even and odd at the origin

a. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = x⁴ - 12x² + 30

Identify the amplitude and period of the following functions.

$f\left(\pi\right)=2\sin2\theta$

Identify the amplitude and period of the following functions.

$g\left(\theta\right)=3\cos\left(\frac{\theta}{3}\right)$

Identify the amplitude and period of the following functions.

$p\left(t\right)=2.5\sin\left(\frac12\left(t-3\right)\right)$

Identify the amplitude and period of the following functions.

$q\left(x\right)=3.6\cos\left(\frac{\pi x}{24}\right)$

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\theta$ (measured in radians) is $A=\frac12r^2\theta$.

<IMAGE>

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

Beginning with the graphs of $y=\sin x$ or $y=\cos x$, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

$q\left(x\right)=3.6\cos\left(\frac{\pi x}{24}\right)+2$

Design a sine function with the given properties.

It has a period of $12$ with a minimum value of $-4$ at $t=0$ and a maximum value of $4$ at $t=6$.

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).