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Functions, Limits, Continuity

Problems for "Brief Review of Functions"

Brief Review of Functions

Limits

Problem : Classify each of the following as a function of x , a one-to-one function, or neither:
a) y = 2x 4 - 6
b) y =
c) y = 4x + 7
d) x = y 2


a) A function of x . Each x is assigned one y . Since each y is assigned to two x values, this is not a one-to-one function:
Figure %:Passes vertical line test. Classified as a function
Figure %:Fails horizontal line test. Not a one-to-one function

b) Also a function of x . Not a one-to-one function
Figure %: y =

c) A one-to-one function. Passes vertical and horizontal line tests.
Figure %: y = 4x + 7

d) Not a function of x . Fails the vertical line test
Figure %: x = y 2

Problem : What is the domain and range of each of the following functions?
a) y = 3sin(2x)
b) y =


a) Domain= (- ∞,∞) ; Range= [- 3, 3]
b) This function can also be expressed as y = . This function looks identical to its reduced form, y = x + 3 , except that it is undefined at x = 3 . Domain= x≠3 ; Range= y≠6
Figure %:Graph of y =

Problem : Let


f (x)   = 4x - 2  
g(x)   = x 2  


a) Find f o g (2)
b) Find g o f (2)


a) f (g(2)) = f (22) = f (4) = 4(4) - 2 = 14
b) g(f (2)) = g(4(2) - 2) = g(6) = 36

Problem : Consider the following piecewise function:

f (x) =    


Find the following:
a) f (- 1)
b) f (5)
c) f (10)


a) f (- 1) = (- 1)3 - 2(- 1) = - 1 - (- 2) = 1
b) f (5) = (5)3 - 2(5) = 125 - 10 = 115
c) f (10) = 17(10) = 170

Problem : Is this function even, odd, or neither?
a) f (x) = cos(x)
b) f (x) = sin(x)
c) f (x) = tan(x)
d) f (x) = cos(x + )


a) Even, cos(- x) = - cos(x)
b) Odd, sin(- x) = - sin(x)
c) Odd, tan(- x) = - tan(x)
d) Neither. Shifting cos(x) to the right by destroys the symmetry with respect to the y -axis.

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