4. Consider the set of functions S = {-3(x² – 4)°,7|x² – 4|³}. a. Show that S is a linearly independent set in C[1,3]. b. Show that S is a linearly dependent set in C[2,3].

Let f: P(N) → P(N) be the f(X) =X∪{8,9} defined function. 1)What is the value when f({6,8})? 2) what is the relationship of f 3)The image of f is? Choose one A) N B) P(N) C) P({8,9}) D) {8,9} E) {X:X∈ P(N) and {8,9}⊆ X} F) All is wrong

Define a relation R from R to R as follows: For all (x, y) ER×R, (x, y) E R if, and only if, x = y² + 1. Prove that (17, -4) ER %3D

1. Determine whether each of the following equations defines y as a function of x. a) x² + y² + 3x=0 b) x-5y=9 c) y = |x-1|

Classify each of the following subsets of R as open, closed, neither, or both. Justify your answer.  (a) (-1, 1) U (10, 11) (b) (-∞, 1] Please include interval notation (with a line) for (a) and (b). Thanks!

10-12 stat

2. Let C = D = {-3, –2, –1, 1, 2, 3} and define a relation S from C to D as follows: For all (x, y) e C x D, 1 (x, y) E S means that 1 is an integer. y a. Is 2 S 2? Is –1S – 1? Is (2, 2) e S? Is (2, –-2) e S? b. Write S as a set of ordered pairs.

3. Consider the functions ƒ₁ (x) = eª, ƒ2(x) = e²¤, and ƒ3 (x) = eª. Call the region of Quadrant I completely enclosed by these three functions by Region R. A. In Quadrant I: functions f₁ and f2 have one intersection point; functions f₁ and f3 have one intersection point; functions f2 and f3 have one intersection point. Find the x-coordinates of these three intersection points. B. Labeling the three x-coordinates from Part A as a < b < c: on the interval [a, b], two of these three functions are the "top" and "bottom" functions defining Region R; on the interval [b, c], two of these three functions are the "top" and "bottom" functions defining Region R. Identify the "top" and "bottom" functions defining Region R on the intervals [a, b] and [b, c]. C. Sketch a graph of Region R.