Functions, Limits, Continuity

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

Solution for Problem 1 >>

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

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

Solution for Problem 2 >>

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 =

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Problem : Let

f o g (2)

g o f (2)

Problem : Consider the following piecewise function:

f (x) =

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 + )

Solution for Problem 5 >>

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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