For each pair of functions give a characteristic that the two functions have in common and a characteristic that distinguishes them.
a) f(x)=1/x and g(x)=sinx
b) f(x)=x and g(x)=[x]
c) f(x)=2^x and g(x)=sinx
a) they are odd
but 1/x is undefined at x = 0
b) linear
but x is odd and |x| is even
c) d/dx (2^x) = 2^x ln 2 so both derivatives keep repeating
but sin x is periodic
a) Both functions, f(x)=1/x and g(x)=sinx, are periodic. However, f(x)=1/x is also asymptotic at x=0, while g(x)=sinx is not.
b) Both functions, f(x)=x and g(x)=[x], are defined for all real numbers. However, f(x)=x is a continuous function, while g(x)=[x] is a step function.
c) Both functions, f(x)=2^x and g(x)=sinx, are oscillatory. However, f(x)=2^x increases exponentially as x increases, while g(x)=sinx oscillates between -1 and 1.
a)
Characteristic in common: Both functions are periodic.
Characteristic distinguishing them: The function f(x)=1/x is not defined at x=0, whereas the function g(x)=sin(x) is defined for all values of x.
b)
Characteristic in common: Both functions are defined for all values of x.
Characteristic distinguishing them: The function f(x)=x is a linear function, whereas the function g(x)=[x] is a step or floor function that takes the greatest integer less than or equal to x.
c)
Characteristic in common: Both functions are defined for all values of x.
Characteristic distinguishing them: The function f(x)=2^x is an exponential function where the values increase rapidly as x increases, whereas the function g(x)=sin(x) is a periodic oscillating function with values between -1 and 1.
a) f(x) = 1/x and g(x) = sin(x):
1) Characteristic in common: Both functions are periodic. The function f(x) = 1/x has a period equal to 2π, while the function g(x) = sin(x) is a periodic function with a period equal to 2π as well.
2) Characteristic that distinguishes them: The range of the two functions is different. The function f(x) = 1/x has a range of all real numbers except for 0, while the function g(x) = sin(x) has a range between -1 and 1.
b) f(x) = x and g(x) = [x]:
1) Characteristic in common: Both functions are non-periodic. The function f(x) = x is a linear function with constant slope, and the function g(x) = [x] represents the greatest integer less than or equal to x.
2) Characteristic that distinguishes them: The functions have different domains and ranges. The function f(x) = x is defined for all real numbers, while g(x) = [x] is only defined for integers. Additionally, the ranges of the functions differ - the function f(x) = x has a range that includes all real numbers, while the function g(x) = [x] has a range of integers only.
c) f(x) = 2^x and g(x) = sin(x):
1) Characteristic in common: Both functions are continuous. The function f(x) = 2^x is continuous for all real numbers, as is the function g(x) = sin(x).
2) Characteristic that distinguishes them: The functions have different rates of growth. The function f(x) = 2^x grows exponentially, while the function g(x) = sin(x) oscillates between -1 and 1. Additionally, the range of the functions differs - the function f(x) = 2^x has a range of positive real numbers, while the function g(x) = sin(x) has a range between -1 and 1.