How do you find the composition of functions gf given fx = 3x + 6 and gx = x + 2, and what is its domain?

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Finding the Composition of Functions gf

Given the functions:

  • f(x) = 3x + 6
  • g(x) = x + 2

We want to find the composition gf, which means we need to substitute f(x) into g(x).

Step 1: Substitute f(x) into g(x)

First, let’s identify the function g with f(x) as input:

g(f(x)) = g(3x + 6)

Now, we replace x in g(x) with 3x + 6:

g(f(x)) = (3x + 6) + 2

This simplifies to:

g(f(x)) = 3x + 8

Step 2: Write Out the Function gf

Thus, the composition of the two functions is:

gf(x) = 3x + 8

Step 3: Determine the Domain of gf

Let’s analyze the domain of gf(x). Since both f(x) and g(x) are defined for all real numbers (the linear functions have no restrictions), the domain of gf(x) is also all real numbers.

In interval notation, the domain can be expressed as:

(−∞, +∞)

Conclusion

The composition function gf(x) is 3x + 8, and its domain is all real numbers, (−∞, +∞).

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