Set Theory, Relations and Functions: Preliminaries: IV:

Problem Set based on previous three parts:

I) Solve the inequalities in the following exercises expressing the solution sets as intervals or unions of intervals. Also, graph each solution set on the real line:

a) $|x| <2$ (b) $|x| \leq 2$ (c) $|t-1| \leq 3$ (d) $|t+2|<1$ (e) $|3y-7|<4$(f) $|2y+5|<1$ (g) $|\frac{z}{5}-1| \leq 1$ (h) $| \frac{3}{2}z-1| \leq 2$ (i) $|3-\frac{1}{x}|<\frac{1}{2}$ (j) $|\frac{2}{x}-4|<3$ (k) $|2x| \geq 4$ (l) $|x+3| \geq \frac{1}{2}$ (m) $|1-x| >1$ (n) $|2-3x| > 5$ (o) $|\frac{x+1}{2}| \geq 1$ (p) $|\frac{3x}{5}-1|>\frac{2}{5}$

II) Quadratic Inequalities:

Solve the inequalities in the following exercises. Express the solution sets as intervals or unions of intervals and graph them. Use the result $\sqrt{a^{2}}=|a|$ as appropriate.

(a) $x^{2}<2$ (b) $4 \leq x^{2}$ (c) $4 (d) $\frac{1}{9} < x^{2} < \frac{1}{4}$ (e) $(x-1)^{2}<4$ (f) $(x+3)^{2}<2$ (g) $x^{2}-x<0$ (h) $x^{2}-x-2 \geq 0$

III) Theory and Examples:

i) Do not fall into the trap $|-a|=a$. For what real numbers a is the equation true? For what real numbers is it false?

ii) Solve the equation: $|x-1|=1-x$.

iii) A proof of the triangle inequality:

Give the reason justifying each of the marked steps in the following proof of the triangle inequality:

$|a+b|^{2}=(a+b)^{2}$…..why ?

$=a^{2}+2ab++b^{2}$

$\leq a^{2}+2|a||b|+b^{2}$….why ?

$\leq |a|^{2}+2|a||b|+|b|^{2}$….why?

$=(|a|+|b|)^{2}$….why ?

iv) Prove that $|ab|=|a||b|$ for any numbers a and b.

v) If $|x| \leq 3$ and $x>-\frac{1}{2}$, what can you say about x?

vi) Graph the inequality: $|x|+|y| \leq 1$

Questions related to functions:

I) Find the domain and range of each function:

a) $f(x)=1-\sqrt{x}$ (b) $F(t)=\frac{1}{1+\sqrt{t}}$ (c) $g(t)=\frac{1}{\sqrt{4-t^{2}}}$

II) Finding formulas for functions:

a) Express the area and perimeter of an equilateral triangle as a function of the triangle’ s side with length s.

b) Express the side length of a square as a function of the cube’s diagonal length d. Then, express the surface area  and volume of the cube as a function of the diagonal length.

c) A point P in the first quadrant lies on the graph of the function $f(x)=\sqrt{x}$. Express the coordinates of P as functions of the slope of the line joining P to the origin.

III) Functions and graphs:

Graph the functions in the questions below. What symmetries, if any, do the graphs have?

a) $y=-x^{3}$ (b) $y=-\frac{1}{x^{2}}$ (c) $y=-\frac{1}{x}$ (d) $y=\frac{1}{|x|}$ (e) $y = \sqrt{|x|}$ (f) $y=\sqrt{-x}$ (g) $y=\frac{x^{3}}{8}$ (h) $y=-4\sqrt{x}$ (i) $y=-x^{\frac{3}{2}}$ (j) $y=(-x)^{\frac{3}{2}}$ (k) $y=(-x)^{\frac{2}{3}}$ (l) $y=-x^{\frac{2}{3}}$

IV) Graph the following equations ad explain why they are not graphs of functions of x. (a) $|y|=x$ (b) $y^{2}=x^{2}$

V) Graph the following equations and explain why they are not graphs of functions of x: (a) $|x|+|y|=1$ (b) $|x+y|=1$

VI) Even and odd functions:

In the following questions, say whether the function is even, odd or neither.

a) $f(x)=3$ (b) $f(x=x^{-5}$ (c) $f(x)=x^{2}+1$ (d) $f(x)=x^{2}+x$ (e) $g(x)=x^{4}+3x^{2}-1$ (f) $g(x)=\frac{1}{x^{2}-1}$ (g) $g(x)=\frac{x}{x^{2}-1}$ (h) $h(t)=\frac{1}{t-1}$ (i) $h(t)=|t^{3}|$ (j) $h(t)=2t+1$ (k) $h(t)=2|t|+1$

Sums, Differences, Products and Quotients:

In the two questions below, find the domains and ranges of $f$, $g$, $f+g$, and $f-g$.

i) $f(x)=x$, $g(x)=\sqrt{x-1}$ (ii) $f(x)=\sqrt{x+1}$, $g(x)=\sqrt{x-1}$

In the two questions below, find the domains and ranges of $f$, $g$, $\frac{f}{g}$ and $\frac{g}{f}$

i) $f(x)=2$, $g(x)=x^{2}+1$

ii) $f(x)=1$, $g(x)=1+\sqrt{x}$

Composites of functions:

1. If $f(x)=x+5$, and $g(x)=x^{2}-5$, find the following: (a) $f(g(0))$ (b) $g(f(0))$ (c) $f(g(x))$ (d) $g(f(x))$ (e) $f(f(-5))$ (f) $g(g(2))$ (g) $f(f(x))$ (h) $g(g(x))$
2. If $f(x)=x-1$ and $g(x)=\frac{1}{x+1}$, find the following: (a) $f(g(\frac{1}{2}))$ (b) $g(f(\frac{1}{2}))$ (c) $f(g(x))$ (d) $g(f(x))$ (e) $f(f(2))$ (f) $g(g(2))$ (g) $f(f(x))$ (h) $g(g(x))$
3. If $u(x)=4x-5$, $v(x)=x^{2}$, and $f(x)=\frac{1}{x}$, find formulas or formulae for the following: (a) $u(v(f(x)))$ (b) $u(f(v(x)))$ (c) $v(u(f(x)))$ (d) $v(f(u(x)))$ (e) $f(u(v(x)))$ (f) $f(v(u(x)))$
4. If $f(x)=\sqrt{x}$, $g(x)=\frac{x}{4}$, and $h(x)=4x-8$, find formulas or formulae for the following: (a) $h(g(f(x)))$ (b) $h(f(g(x)))$ (c) $g(h(f(x)))$ (d) $g(f(h(x)))$ (e) $f(g(h(x)))$ (f) $f(h(g(x)))$

Let $f(x)=x-5$, $g(x)=\sqrt{x}$, $h(x)=x^{3}$, and $f(x)=2x$. Express each of the functions in the questions below as a composite involving one or more of f, g, h and j:

a) $y=\sqrt{x}-3$ (b) $y=2\sqrt{x}$ (c) $y=x^{\frac{1}{4}}$ (d) $y=4x$ (e) $y=\sqrt{(x-3)^{3}}$ (f) $y=(2x-6)^{3}$ (g) $y=2x-3$ (h) $y=x^{\frac{3}{2}}$ (i) $y=x^{9}$ (k) $y=x-6$ (l) $y=2\sqrt{x-3}$ (m) $\sqrt{x^{3}-3}$

Questions:

a) $g(x)=x-7$, $f(x)=\sqrt{x}$, find $(f \circ g)(x)$

b) $g(x)=x+2$, $f(x)=3x$, find $(f \circ g)(x)$

c) $f(x)=\sqrt{x-5}$, $(f \circ g)(x)=\sqrt{x^{2}-5}$, find $g(x)$.

d) $f(x)=\frac{x}{x-1}$, $g(x)=\frac{x}{x-1}$, find $(f \circ g)(x)$

e) $f(x)=1+\frac{1}{x}$, $(f \circ g)(x)=x$, find $g(x)$.

f) $g(x)=\frac{1}{x}$, $(f \circ g)(x)=x$, find $f(x)$.

Reference: Calculus and Analytic Geometry, G B Thomas.

NB: I have used an old edition (printed version) to prepare the above. The latest edition may be found at Amazon India link:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=1XDE2XDSY5LCP&keywords=gb+thomas+calculus&qid=1570492794&s=books&sprefix=G+B+Th%2Caps%2C255&sr=1-1

Regards,

Nalin Pithwa

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