$$\begin{array} { l } { \text { Prove that isomorphism is an equivalence relation. That is, for any } } \\ { \text { groups } G , H , \text { and } K , G \approx G , G \approx H \text { implies } H \approx G , \text { and } G \approx H \text { and } } \\ { H \approx K \text { implies } G \approx K } \end{array}$$

Find the y-coordinate corresponding to each x-coordinate if the functions are f(x) = -2x - 5 and g(x) = 3.75(2.5)^x. Check your answers with your calculator. a. f(6) b. f(0) c. g(2) d. g(-2) e. f(-4) f. g(-1) g. f(3) h. g(0)

Suppose that functions ƒ and g are continuous on [a, b] and differentiable throughout (a, b) and suppose also that g' ≠ 0 throughout (a, b). Then there exist a number c in (a, b) at which ƒ'(c)/g'(c) = ƒ(b)-ƒ(a)/g(b)-g(a). Find all values of c in (a, b) that satisfy this property for the following given functions and intervals. ƒ(x) = x³+1, g(x) = x²-x, [a, b] = [-1, 1]. ƒ(x) = cos x, g(x) = sin x, [a, b] = [0, π/2]

For the pair of functions, (a) find $(f+g)(x),(f-g)(x)$, and $(f g)(x)$; (b) give the domains of the functions in part (a); (c) find $\frac{f}{g}$ and give its domain; (d) find $f \circ g$ and give its domain; and (e) find $g \circ f$ and give its domain. Do not use a calculator.

$$f(x)=\sqrt[3]{x+4}, g(x)=x^3+5$$

f(x) = 2x^2 - 3 for x is greater than or equal to 0.

(a) Is f(x) one-to-one?

(b) If g(x) is the inverse of f(x), use the formula g'(x)=1/(f'(g(x))) to find g'(0).

(c) Find an algebraic expression for g(x) and use it to confirm the answer in part (b).

(d) Sketch the graphs of f(x) and g(x). Then use the calculator to confirm the value of g'(0) found in part (b).

For which pair(s) of functions is $[f \circ g](x)$ a quadratic function? A. $f(x)=x^{2} \text { and } g(x)=3 x+7$ B. $f(x)=x-4 \text { and } g(x)=x^{2}-1$ C. $f(x)=x+9 \text { and } g(x)=2 x-3$ D. $f(x)=x \text { and } g(x)=6 x$ E. $f(x)=x^{2}-1 \text { and } g(x)=x^{2}+1$ F. $f(x)=x^{2}-4 \text { and } g(x)=3 x$

Use a graphing calculator to evaluate

$$( f + g ) ( x ) , ( f - g ) ( x ) , ( f g ) ( x ) , \text { and } \left( \frac { f } { g } \right) ( x )$$

when x = 5. Round your answers to two decimal places.

$$f ( x ) = 4 x ^ { 4 } ; g ( x ) = 24 x ^ { 1 / 3 }$$

Let $\varphi$ : G $\rightarrow$ GL(V) be a representation with character $\psi$. Let W be the subspace

$$\{v \in V | \varphi(g)(v)=v$$

for all g $\in$ G} of V fixed pointwise by all elements of G. Prove that dim W = $\left(\psi, \chi_{1}\right),$ where $\chi_{1}$ is the principal character of G.

Write the equilibrium expression for each of the following reactions.

a. $\mathrm{CO}(\mathrm{g})+2 \mathrm{H}_2(\mathrm{~g}) \rightleftharpoons \mathrm{CH}_3 \mathrm{OH}(\mathrm{g})$
b. $2 \mathrm{NO}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g})$
c. $\mathrm{P}_4(g)+6 \mathrm{Br}_2(g) \rightleftharpoons 4 \mathrm{PBr}_3(g)$

Calculate the mass percent composition of each of the following: a. $0.890 \mathrm{~g}$ of $\mathrm{Ba}$ and $1.04 \mathrm{~g}$ of $\mathrm{Br}$ b. $3.82 \mathrm{~g}$ of $\mathrm{K}$ and $1.18 \mathrm{~g}$ of $\mathrm{I}$ c. $3.29$ of N, $0.946 \mathrm{~g}$ of $\mathrm{H}$, and $3.76 \mathrm{~g}$ of $\mathrm{S}$ d. $4.14 \mathrm{~g}$ of $\mathrm{C}, 0.695 \mathrm{~g}$ of $\mathrm{H}$, and $2.76 \mathrm{~g}$ of $\mathrm{O}$

Let G denote a group, and H a subgroup of G. Prove:\

If a is any element of $G,(\langle a\rangle)$ is a normal subgroup of G iff a has the following property: For any $x \in$ G, there is a positive integer k such that $x a=a^k x$.

The square of g is the same as two times the difference of g and 10. Match the sentence with an equation. A.

$$g^2 = 2(g - 10)$$

B.

$$\frac{1}{2}g + 32 = 15 + 6g$$

C.

$$g^3 = 24g + 4$$

D.

$$3g^2 = 30 + 9g$$

Use $\triangle A P G$ and $\triangle G R A$ in the diagram from Question mentioned to prove that opposite angles of a parallelogram are congruent. Create a proof of the statement $\angle G P A \cong \angle A R G$.

Given: Parallelogram PARG with diagonals $\overline{P R}$ and $\overline{A G}$ intersecting at point M

Prove: $\angle G P A \cong \angle A R G$

What mass of a $1.0$-g sample of radon remains after $30$ days? (The half-life of radon is $3.8$ days.)

(a) $0.030 \mathrm{~g}$

(b) $0.0040 \mathrm{~g}$

(c) $0.99 \mathrm{~g}$

(d) $0.97 \mathrm{~g}$

(e) $0.0010 \mathrm{~g}$