Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 3/(x-4) and g(x) = (3/x) + 4

Verified step by step guidance

1. First, recall that the composition of functions f(g(x)) means substituting g(x) into every x in f(x). So, write down f(g(x)) as f\left(g(x)\right) = f\left(\frac{3}{x} + 4\right).
2. Next, substitute g(x) = \frac{3}{x} + 4 into f(x) = \frac{3}{x - 4}. This gives f(g(x)) = \frac{3}{\left(\frac{3}{x} + 4\right) - 4}.
3. Simplify the denominator of f(g(x)) by combining like terms inside the parentheses: \left(\frac{3}{x} + 4\right) - 4 = \frac{3}{x}. So, f(g(x)) = \frac{3}{\frac{3}{x}}.
4. Now, simplify the complex fraction \frac{3}{\frac{3}{x}} by multiplying numerator and denominator appropriately, which will simplify to x.
5. Repeat the process for g(f(x)): substitute f(x) into g(x), so g(f(x)) = g\left(\frac{3}{x - 4}\right) = \frac{3}{\frac{3}{x - 4}} + 4, then simplify this expression step-by-step. Finally, check if both compositions simplify to x, which would indicate that f and g are inverses.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to analyze combined transformations or operations.

Inverse Functions

Two functions f and g are inverses if composing them in either order returns the input, meaning f(g(x)) = x and g(f(x)) = x. This relationship shows that each function reverses the effect of the other, effectively undoing the transformation.

Rational Functions and Domain Restrictions

Rational functions are ratios of polynomials and may have restrictions where the denominator is zero. Understanding these domain restrictions is crucial when composing functions or checking inverses to avoid undefined expressions and ensure valid operations.