To determine whether the functions f and g are inverses, we need to verify that the compositions fg(x) and gf(x) equal x. In other words, we will check if:
- fg(x) = x
- gf(x) = x
Given the functions:
f(x) = 7x + 3
g(x) = 3x – 7
Step 1: Calculate fg(x)
First, we substitute g(x) into f:
fg(x) = f(g(x)) = f(3x – 7)
Now, we will substitute (3x – 7) into f(x):
f(3x – 7) = 7(3x – 7) + 3
Expanding this gives:
= 21x – 49 + 3
= 21x – 46
So, we find:
fg(x) = 21x – 46
However, we need this to equal x for the functions to be inverses. It appears that we have made an error in substituting; let’s simplify this process in a direct manner.
Step 2: Check if gf(x)
Next, we calculate gf(x):
gf(x) = g(f(x)) = g(7x + 3)
Now substituting into g(x):
g(7x + 3) = 3(7x + 3) – 7
Distributing gives:
= 21x + 9 – 7
= 21x + 2
Again, this does not yield x directly, which suggests a need for value manipulation or further inspection of our initial functions.
Final Assessment
After reviewing these compositions:
We have fg(x) = 21x – 46 and gf(x) = 21x + 2. Neither of these compositions yield x, indicating that the functions f(x) = 7x + 3 and g(x) = 3x – 7 are not inverses of each other.
In conclusion, the functions do not fulfill the requirement for being inverses, and further evaluation indicates that their outputs do not coincide with the identity function as anticipated.