⧼exchistory⧽
11 exercise(s) shown, 0 hidden
BBy Bot
Nov 02'24
[math]
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[/math]
Plot the following point in the [math]xy[/math]-plane: [math](0, -2)[/math], [math](1, 3)[/math], [math](3, 1)[/math], [math](-4, -4)[/math], and [math](5, 0)[/math].
BBy Bot
Nov 02'24
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[/math]
In the [math]xy[/math]-plane plot the points [math](1, 2)[/math] and [math](2, 1)[/math], [math](-3, 2)[/math] and [math](2, -3)[/math], [math](-2, -3)[/math] and [math](-3, -2)[/math]. Describe the relative positions of the points [math](a, b)[/math] and [math](b, a)[/math] for arbitrary [math]a[/math] and [math]b[/math].
BBy Bot
Nov 02'24
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[/math]
The [math]x[/math]-axis and the [math]y[/math]-axis divide [math]\R^2[/math] into four quadrants, as shown in Figure. Let [math](a, b)[/math] be a point for which neither [math]a[/math] nor [math]b[/math] is zero. How can you recognize instantly which quadrant [math](a, b)[/math] belongs to?
BBy Bot
Nov 03'24
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[/math]
Find the distance between [math](-1, 2)[/math] and [math](3, 4)[/math]; [math](2,3)[/math] and [math](3,2)[/math]; [math](3,4)[/math] and [math](-1,2)[/math]; [math](-2,1)[/math] and [math](2,1)[/math]. In each case plot the points in [math]\R^2[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
Verify Proposition.
BBy Bot
Nov 03'24
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[/math]
Plot the subsets of the [math]xy[/math]-plane defined in (i) through (vi).
BBy Bot
Nov 03'24
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[/math]
In each of the following, plot the subset of [math]\R^2[/math] that consists of all pairs [math](x, y)[/math] such that the given equation (or conditions) is satisfied.
- [math]3x + 2y = 3[/math]
- [math]x + y = 1[/math]
- [math]y = |x|[/math]
- [math]y = \sqrt x[/math]
- [math]x^2 + y^2 = 4[/math]
- [math]x^2 + 4y^2 = 4[/math]
- [math]x^2 + y^2 = 1[/math] and [math]y \geq 0[/math]
- [math]4x^2 - y^2 = 4[/math]
- [math]y = 2x^2 + x - 2[/math]
- [math]y = |x^3|[/math]
- [math]y = \mbox{largest integer less than or equal to}\ x[/math]
- [math]y = \dilemma{2x + 3, & x \geq 0}{\frac{x^2}{2}, & x \lt 0.}[/math]
BBy Bot
Nov 03'24
[math]
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[/math]
In Problem Exercise, which subsets are functions?
BBy Bot
Nov 03'24
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[/math]
Let [math]f[/math] and [math]g[/math] be two functions defined, respectively, by
[[math]]
\cond{f(x)= x^2 + x + 1, & -\infty \lt x \lt \infty}
,
[[/math]]
[[math]]
\cond{g(x)= \frac{x + 1}{x - 1}, & \mbox{for every real number $x$ except \ltmath\gtx=1[[/math]]
}} . </math> Find:
- [math]f(2)[/math], [math]f(0)[/math], [math]f(a)[/math], [math]f(a + b)[/math], [math]f(a - b)[/math].
- [math]g(0)[/math], [math]g(-1)[/math], [math]g(10)[/math], [math]g(5 + t)[/math], [math]g(x^3)[/math].
BBy Bot
Nov 03'24
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[/math]
Give an example of a function [math]f[/math] and a function [math]g[/math] that satisfy each of the following conditions.
- domain [math]f =[/math] domain [math]g[/math], but range [math]f \ne[/math] range g.
- domain [math]f \ne[/math] domain [math]g[/math], but range [math]f =[/math] range g.
- domain [math]f =[/math] domain [math]g[/math] and range [math]f =[/math] range [math]g[/math], but [math]f \ne g[/math].
- [math]f(a) = g(a)[/math] for every [math]a[/math] that belongs to both domains, but [math]f \ne g[/math].