what is the difference between the function 2^x and (1/2)^x?
well, (1/2) = 2^-1
go to wolframalpha.com and enter
plot y=2^x, y=(1/2)^x
Oh ok! so 2^x is the inverse of (1/2)^x
the difference between the function 2^x and (1/2)^x
= |2^x - (1/2)^x|
= | 2^x - 2^-x |
Oh I wrote the question incorrect. It's not to find the difference but to explain the difference between the two functions
then go with Steve's suggestion and plot the two curves
y = 2^x and y=(1/2)^x to see the result
http://www.wolframalpha.com/input/?i=plot+y+%3D+2%5Ex+%2C+y+%3D+%281%2F2%29%5Ex
2^x is not the inverse of (1/2)^x
log_2(x) is the inverse of 2^x
log_1/2(x) is the inverse of (1/2)^x
You can check this because if g(x) is the inverse of f(x), f(g(x)) = x and g(f(x)) = x
(1/2)^x = 1/2^x
they are reciprocals, not inverses.
To understand the difference between the functions 2^x and (1/2)^x, we need to look at the properties of exponents and how they affect these functions.
Both functions involve raising a number to the power of x, where x is a variable.
1. 2^x:
When we have 2^x, we are raising the number 2 to the power of x. This means that we multiply 2 by itself x times.
For example:
2^3 = 2*2*2 = 8
2^2 = 2*2 = 4
2^1 = 2
As x increases, the value of 2^x grows rapidly. For positive values of x, the function 2^x represents exponential growth. This means that as x increases, the function value increases exponentially (at an accelerated rate).
2. (1/2)^x:
When we have (1/2)^x, we are raising the number 1/2 (which is equivalent to 0.5) to the power of x. This means that we multiply 1/2 by itself x times.
For example:
(1/2)^3 = (1/2)*(1/2)*(1/2) = 1/8
(1/2)^2 = (1/2)*(1/2) = 1/4
(1/2)^1 = 1/2
As x increases, the value of (1/2)^x decreases rapidly. For positive values of x, the function (1/2)^x represents exponential decay. This means that as x increases, the function value decreases exponentially (at a decelerated rate).
In summary, the main difference between the functions 2^x and (1/2)^x is their behavior as x increases. 2^x represents exponential growth, while (1/2)^x represents exponential decay.