Mathematics Magazine for Grades 112 





Theory:An identical Equation
(identity) is one in which allpossible values of the variable will satisfy the
equation. Example of identities x + x = 2x. No mater what real numbers are
substituted for x and y Left side = Right Side of the equation. A conditional Equation is
satisfied by only certain values of the variable. For instance x + 2 = 5 is
satisfied only by x = 3; x + xy = 4 is satisfied by (2,1) and other pairs also,
but for most pairs of Reals Left side ≠ Right Side of the equation. Solutions from the Previous Issue: 1. If f(x) = x^{2 }and g(x) = x – 3 find the composite functions f ^{o} g and g ^{o} f and their domains. Solution: (f ^{o} g)(x) = f(g(x)) = f(x  3) = (x  3)^{2} (g ^{o} f)(x) =
g(f(x)) = g(x^{2}) = x^{2}  3 The domain of both f and g are R (the set of all
real numbers) 2. If and find the function f ^{o} g, g ^{o} f and their domains. Solution: (f ^{o} g)(x) = f(g(x)) = = f(
) =
The domain of f ^{o} g is {x 2 – x ≥ 0} = {x  x ≤ 2} = (∞, 2]. (g ^{o} f
)(x) = g(f(x)) = g(
) =
For
to be defined we must have x ≥ 0. For
to be defined we must have
≥ 0, that is
or x ≤ 4. Thus we have 0 ≤ x ≤ 4so the domain of g ^{o}
f is the closed interval [0,
4] 3. Given find the functions f and g such that F= f ^{o} g. Solution: Since the formula for F says to first add 9 and then take the fourth root , we let g(x) = x + 9 and f(x) = Then (f ^{o} g)(x)= f(g(x)) = f(x + 9) = = F(x) Proposed Problems: 1. Can you draw an equilateral triangle to fit snugly inside a square, so that each of its three vertices touches one of the sides of the square? Is there only one such triangle? If not, what is the smallest and biggest equilateral triangle you can draw to fit snugly within a square? 2. Reconstruct a triangle from three points that you know to be the midpoints of its three sides. What if you only know the three points to be the midpoints of the triangle’s medians? (A ‘median’ is the line joining a vertex of the triangle with the centre of the side opposite) 3. Anyone can draw a square inside an equilateral triangle. But what is the biggest square you can draw inside a 1m equilateral triangle, and how do you position it? Are there any 'competing' ways to position the square? 

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