How can we show that (f o g)(g o f)(x) for the functions f(x) = 2x + 1 and g(x) = x + 12?

by

To demonstrate that (f o g)(g o f)(x) holds true for the functions f(x) = 2x + 1 and g(x) = x + 12, we first need to understand what the composition of functions means.

The notation f o g represents the function f applied to the output of the function g. Similarly, g o f represents the function g applied to the output of the function f.

Step 1: Calculate f(g(x))

First, we plug g(x) into f(x):

  • Replacing x in f(x): f(g(x)) = f(x + 12)
  • Substituting: f(x + 12) = 2(x + 12) + 1
  • Expanding: 2x + 24 + 1 = 2x + 25

Thus, f(g(x)) = 2x + 25.

Step 2: Calculate g(f(x))

Now, we compute g(f(x)):

  • Replacing x in g(x): g(f(x)) = g(2x + 1)
  • Substituting: g(2x + 1) = (2x + 1) + 12
  • Simplifying: 2x + 1 + 12 = 2x + 13

Thus, g(f(x)) = 2x + 13.

Step 3: Calculate (f o g)(g o f)(x)

We have now determined:

  • f(g(x)) = 2x + 25
  • g(f(x)) = 2x + 13

We can evaluate the composition as follows:

  • (f o g)(g o f)(x) = f(g(f(g(x))))
  • Substituting: = f(g(2x + 13)) = f(2x + 13 + 12)
  • Calculating: f(2x + 25) = 2(2x + 25) + 1 = 4x + 50 + 1 = 4x + 51

Conclusion

Thus, we have demonstrated that the compositions of the functions give us a determinate result. Specifically, we found that:

(f o g)(g o f)(x) = 4x + 51.

Leave a Comment