(Solved): 24. Give an example of an increasing function whose domain is the interval \( [0,1] \) but whose ra ...

24. Give an example of an increasing function whose domain is the interval \( [0,1] \) but whose range does not equal the interval [f(0), (1)]. 25. Show that the sum of two increasing functions is increasing. 26. Give an example of two increasing functions whose product is not increasing. [Hint: There are no such examples where both functions are positive everywhere.] 27. Give an example of two decreasing functions whose product is increasing. 28. Show that the compositioa of two increasing functions is increasing. 29. Show that the composition of two linear functions is a linear function. 30. Show that if fand \( g \) are linear functions, then the graphs of \( f * g \) and \( g * f \) have the same slope: 31. Show that if \( f \) is a non-constant linear function and \( g \) is a quadratic function, then \( f \circ g \) and \( g \circ f \) are both quadratic functioni. 12. Suppose \( f \) is a quadratic function such that the equation \( f(x)=0 \) has exactly one solution. Show that this solution is the firt. coortinate of the vertex of the graph of \( f \) and that the second coordinate of the vertex equals 0 . 33. Suppose \( f \) is a quadntic function such that the equation \( f(x)=0 \) has two real solutions. Show that the average of these two solutions is the fint coondinate of the vertex of the graph of \( f \) : 34. Show that \( \cos \frac{\pi}{32}=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2} \) [Hint: Fint do Exercise 66] 35. Suppose \( u=\arctan 2 \) and \( v=\arctan 3 \). Show that \( \tan (u+v)=-1 \). 36. Suppose that \( u=\arctan 2 \) and \( v=\arctan 3 \). Using the previous problem, explain why \( +v=\frac{\text { in }}{4} \). 37. Use the previous problem to derive the beautifil equation \( \arctan 1+\arctan 2+\arctan 3=\pi \).