Use a graphing utility to determine whether the proposed identity appears to be an identity. If it does, prove it algebraically. If not, give a counterexample.

$\cos 3 x=3 \cos x$

Prove the statement given below. Write your answer.

$\forall$ real numbers $r$ and $\mathcal{I}$, if $r$ is rational and $\mathcal{I}$ is irrational, then $\frac{r}{\mathcal{I}}$ is irrational.

$\dfrac{a}{b}+b$. Variable represent real numbers. a) Write each expression as a single rational expression in lowest terms b) Indicate the domain of the expression c) Check with a CAS or by substitution

For this exercise, you will need to prove that $\sqrt{\frac{2-\sqrt{3}}{4}}=\frac{\sqrt{6}-\sqrt{2}}{4}$ in the following way.

a. Find a value for $\sin \frac{\pi}{12}$ by thinking of $\frac{\pi}{12}$ as $\frac{\pi}{3}-\frac{\pi}{4}$ and using the identity for $\sin (\alpha-\beta)$.

b. Find a value for $\sin \frac{\pi}{12}$ by thinking of $\frac{\pi}{6}$ as twice $\frac{\pi}{12}$ and using an identity for $\cos 2 x$.

c. Why do Parts a and b prove the desired result?

We known a graph of $y=\tan^2 x$. Use the identity $\tan^2 x+1=\sec^2 x$ to sketch a graph of $y=\sec^2 x$

Write as a simple fraction and state any restrictions on the variables.

$$\frac{\frac{1}{(2+h)^2}-\frac{1}{4}}{h}$$

Rewrite the rational expression in the lowest terms, and identify the domain for both the original expression and the form in the lowest terms.

$\frac{b^3-5 b^2-6 b}{b^2-b-2}$

Simplify given expression $\dfrac{\dfrac{(x+1)^2}{x}}{\dfrac{x^2-1}{x^2}}$and give all restrictions on the varable

An elegant alternate proof of two of the identities of this lesson is possible using matrices. For any real number$\theta$,

$$\begin{bmatrix} \cos\theta & -\sin\theta \\\sin\theta & \cos\theta \end{bmatrix}$$

is the matrix for $R_{\theta}$, where $R_{\theta}$ is the rotation of $\theta$ about the origin. Translate the transformacion identity $R_{\theta+\phi}=R_{\theta} \circ R_{\phi}$ into matrices and then multiply the matrices. Which two identities result?

For this exercise, you will need to graph the function f defined by $f(x)=-\frac{6}{(2 x-3)^2}$ and write equations for its horizontal and vertical asymptotes.

a. Sketch a graph of the function.

b. For what value(s) is the function undefined?

c. Identify any discontinuities as essential or removable.

d. Find an equation that describes the end behavior.

$$h: t \rightarrow \frac{t-2}{t^2-2 t}$$

a. Sketch a graph of the function.

b. For what value(s) is the function undefined?

c. Identify any discontinuities as essential or removable.

d. Find an equation that describes the end behavior.

$$g: x \rightarrow \frac{3 x^2-3 x}{4 x^2-1}$$

a. Sketch a graph of the function.

b. For what value(s) is the function undefined?

c. Identify any discontinuities as essential or removable.

d. Find an equation that describes the end behavior.

$$f: x \rightarrow \frac{x^3+3 x^2-10}{x+4}$$

Consider a function f with the following properties:

$$\begin{array}{ll} \lim _{x \rightarrow 2^{+}} f(x)=-\infty & \lim _{x \rightarrow 2^{-}} f(x)=-\infty \\ \lim _{x \rightarrow \infty} f(x)=3 & \lim _{x \rightarrow-\infty} f(x)=3 \end{array}$$

a. Construct a possible graph of function f.

b. Write equations for its horizontal and vertical asymptotes.