Tag Archives: triangle inequality proof

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<x^{2}<9 (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