Explain why or why not. Determine whether the following statements are true and give an explanation or counter example. a. A function could have the property that ?f?(x ? ?) =? ?(? ? for all ? . b. cos (a? ? +? ?)= cos ?a? + cos ?b? for all? ? and ?b? in [0, 2?? . c. If ?f? is a linear function of the form ?f?(?x?) = ?mx? + ?b?,then ?f?(?u? + ?v?)= ?f?(?u?)+ ?f?(?v?)for all ?u and v? . d. The function f ? ?(? ? 1 ? ? ? has the property ?f(f(x)) = x e. The set {?x?: | ?x? + 3| > 4} can be drawn on the number line without lifting your pencil. f. logic (x? y?) = (logl0 ? ? (logl0 ?y?). g. sin?1 (sin ?(2??))= 0.

Step-by-step solution Step 1 a) We need to find whether the statement “A function could have the property that f(x) = f(x) for all x” is true or false. Step 2 By the definition of an even function, we know that a function f is even if f satisfies f(x) = f(x) for every number x in its domain. With this, th graph of f is symmetric with respect to the y-axis. Step 3 For example, the function f(x) = x is even because f(x) = (x) = x = f(x). Step 4 Graphing our example to determine the symmetry, we get: Step 5 Therefore, the statement “A function could have the property that f(x) = f(x) for all ” is true if f is an even function. Step 6 b) We need to find whether thestatement “cos (a+b) = cos a + cos b for all a and b in [0, 2]” is true or false. Step 7 To solve theproblem, we use an example. We assign values for a and b interval [0, 2] and evaluate cos (a+b) and cos a + cos b . Step 8 Let a = 2 and b = 32 , then, 3 cos ( 2 ) 2 1 cos 2 +cos 32 = 0 3 3 cos ( +2) c2s 2 +cos 2 Step 9 Th erefor e, the statement “cos (a+b) = cos a + cos b for all a and b in [0, 2]” is false. Step 10 c) We need to find whether the statement “If f is a linear function of the form f(x) = mx+b, then f(u+v) = f(u)+f(v) for all u and v” is true or false. Step 11 Given that f is a linear function and f(x) = mx+b, then, f(u+v) = m(u+v)+b [1] Since b is not dependent on the variable, [1] becomes f(u+v) = m(u+v) [2] Step 12 Next, we get an expression for f(u)+f(v) f(u)+f(v) = mu+mv [3] Simplifying [3], we get: f(u)+f(v) = m(u+v) [4] Step 13 From equations [2] and [4], we see that f(u+v) = f(u)+f(v) Step 14 Therefore, the statement “If f is a linear function of the form f(x) = mx+b, then f(u+v) = f(u)+f(v) for all u and v” is true. Step 15 d) We need to find whether the statement “The function f(x) = 1x has the property f(f(x)) = x” is true or false.