Determine algebraically whether f(x)=3x-2 and g(x)= x+2/3 are
inverse functions.
We can determine if two functions, f and g, are inverse functions by checking if the composition of f and g and the composition of g and f is the identity function, f(g(x)) = x and g(f(x)) = x.
First, we check the composition f(g(x)):
f(g(x)) = f(x + 2/3)
= 3(x + 2/3) - 2
= 3x + 2 - 2
= 3x
Since f(g(x)) = 3x, which is not equal to x, f and g are not inverse functions.
Second, we check the composition g(f(x)):
g(f(x)) = g(3x - 2)
= 3x - 2 + 2/3
= 3x - 2/3
Since g(f(x)) = 3x - 2/3, which is not equal to x, g and f are not inverse functions.
Therefore, f(x) = 3x - 2 and g(x) = x + 2/3 are not inverse functions.
Rewrite the following set: {x|-11 ≤ x < 8} in interval notation.
The set {x|-11 ≤ x < 8} can be rewritten in interval notation as [-11, 8).
To determine if two functions, f(x) and g(x), are inverse functions algebraically, we need to show that the composition of the two functions equals the identity function.
The composition of f(g(x)) is as follows:
f(g(x)) = f(x + 2/3) = 3(x + 2/3) - 2 = 3x + 2 - 2 = 3x
If f(g(x)) = 3x, then g(x) is the inverse of f(x).
Now we need to check the other composition, g(f(x)):
g(f(x)) = g(3x - 2) = (3x - 2) + 2/3
Simplifying further:
g(f(x)) = 3x - 2 + 2/3 = 3x - (6/3) + 2/3 = 3x - 4/3
Since g(f(x)) is not equal to x, the two functions, f(x) = 3x - 2 and g(x) = x + 2/3, are not inverse functions.
Therefore, f(x) and g(x) are not inverse functions algebraically.