1. The following functions are inverses of each other: f(x) = 3x - 7 and g(x)= x+3 / 7
False, right?
To determine whether the functions f(x) = 3x - 7 and g(x) = (x+3) / 7 are inverses of each other, we need to check if performing the two functions in succession will lead to the original value of x.
First, let's find the composite function f(g(x)). To do this, we substitute g(x) into f(x) and simplify:
f(g(x)) = f((x+3)/7)
= 3((x+3)/7) - 7
= (3x + 9) / 7 - 7
= (3x + 9 - 49) / 7
= (3x - 40) / 7
Now let's find the composite function g(f(x)):
g(f(x)) = g(3x - 7)
= (3x - 7 + 3) / 7
= (3x - 4) / 7
If f(x) and g(x) are inverses of each other, then f(g(x)) = g(f(x)) = x.
Comparing the two composite functions, we can see that (3x - 40) / 7 is not equal to (3x - 4) / 7. Therefore, the given functions f(x) = 3x - 7 and g(x) = (x+3) / 7 are not inverses of each other.
So, the answer is true, not false.
if they are inverses, then f(g(x)) = g(f(x) = 1
So, what do we have?
f(g) = 3g-7 = 3((x+3)/7)-7 ≠ x
Note that if g = (x+7)/3, then f = g-1