More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


ƒ o G

Verified step by step guidance

1. Identify the functions involved: \( f(x) = |x| \) and \( G(x) = \frac{1}{x-2} \).
2. Understand the composition \( (f \circ G)(x) = f(G(x)) \), which means substituting \( G(x) \) into \( f(x) \).
3. Substitute \( G(x) = \frac{1}{x-2} \) into \( f(x) \), resulting in \( f\left(\frac{1}{x-2}\right) = \left| \frac{1}{x-2} \right| \).
4. Determine the domain of \( G(x) = \frac{1}{x-2} \), which is all real numbers except \( x = 2 \) because the denominator cannot be zero.
5. Since \( f(x) = |x| \) is defined for all real numbers, the domain of \( f \circ G \) is the same as the domain of \( G(x) \), which is all real numbers except \( x = 2 \).

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (f o g)(x) = f(g(x)). Understanding how to combine functions is essential for solving problems involving multiple functions, as it requires evaluating the inner function first before applying the outer function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with composite functions, it is crucial to determine the domain of both the inner and outer functions, as the overall domain will be restricted by any values that make either function undefined.

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x regardless of its sign. This function is important in composite functions because it affects the output based on the input's sign, which can influence the overall behavior and domain of the composite function.