Determine whether the statement is true or false?

a)If x1 %26lt; x2 and f is a decreasing function, then f(x1) %26gt; f(x2).





b) If f and g are functions, then f o g = g o f.





c) If f is a function, then f(s + t) = f(s) + f(t)





d) If f(s) = f(t), then s = t.





d) If f is a function, then f(3x) = 3f(x).Determine whether the statement is true or false?
All of the statements are false (except possibly a, see below):





a) The definition of a decreasing function is: f is decreasing if whenever x_1 %26lt; x_2, f(x_1) 鈮?f(x_2). So in particular, f(x_1) might be equal to f(x_2) and thus not strictly greater than it. So this statement is false, according to the above. Note, however, that some authors use decreasing to mean strictly decreasing, in which case the ';鈮?quot; in the definition above is replaced with ';%26gt;';, thus rendering this statement true. Check to be sure which convention your textbook uses.





b) Absolutely false. Consider the following -- let f(x) = 2x and g(x) = x+1. Then we have (f鈭榞)(x) = 2(x+1) = 2x+2 and (g鈭榝)(x) = 2x+1 鈮?(f鈭榞)(x). So function composition is not commutative.





c) False. Consider, for instance, f(x) = x虏. Then f(1+1) = 4 鈮?1+1 = f(1) + f(1).





d) False. Again, consider f(x) = x虏. Then f(-1) = f(1) = 1, but -1 鈮?1.





e) False. Again, consider f(x) = x虏. Then f(3x) = 9f(x) 鈮?f(x) (unless x=0, which it usually isn't).Determine whether the statement is true or false?
t


f


f


f


f

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