Functions, Limits, Continuity: Problems for "Brief Review of Functions"

Problem : Classify each of the following as a function of x, a one-to-one function, or neither:
a) y = 2x⁴ - 6
b) y =
c) y = 4x + 7
d) x = y²

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:

b) Also a function of x. Not a one-to-one function

c) A one-to-one function. Passes vertical and horizontal line tests.

d) Not a function of x. Fails the vertical line test

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

Problem : Let

f (x)		= 4x - 2
g(x)		= x²

a) Find

fog

(2)
b) Find

gof

(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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