g(f(4)) = g(10) = 10^2 + 1 = 100 + 1 = 101. - Appfinity Technologies

Understanding the Context

What Are Functions and Compositions?

Before diving into the calculation, let’s clarify the basic concepts. A function assigns a unique output to each input value. When we write g(f(x)), we compute the inner function f(x) first, then apply g to that result — a process called function composition:
g(f(x)) means “g evaluated at the value of f(x).”

Tracing g(f(4))

Key Insights

To understand g(f(4)), we need two things:

1. The value of f(4)
   
2. The function g defined such that g(10) = 101, and g(10) arises from f(4)

The equation g(f(4)) = g(10) = 101 tells us that f(4) must equal 10. Why? Because if g(10) = 101, then inputting 10 into g yields 101. For g(f(4)) to equal 101, f(4) must be 10 — this is the core principle of function evaluation.

Exploring Possible Definitions of f and g

While the exact definitions of f and g aren't fixed, they are constrained by the equation:

Final Thoughts

• f(4) = 10
  This fixes one key input-output pair.
  
• g(10) = 101
  This defines the behavior of g at input 10.

One simple way to interpret this is that g(x) = x² + 1
Then:

• g(10) = 10² + 1 = 100 + 1 = 101
  
• Since f(4) = 10, substituting:
  g(f(4)) = g(10) = 101

This aligns perfectly with the given identity.

Is g(x) = x² + 1 the Only Possibility?

No — the expression holds broadly across many functions. For example:

• If g(x) = 10x + 1, then g(10) = 10×10 + 1 = 101 still holds.
  
• Any function g satisfying g(10) = 101 will validate g(f(4)) = 101 when f(4) = 10.
  But given the simplicity of the result, g(x) = x² + 1 is a natural and elegant choice.