91Ó°ÊÓ

Understanding the domain and range of a function is crucial for fully grasping function compositions. The **domain** of a function is the set of all possible input values (the x-values) which will produce a valid output, while the **range** is the set of all possible output values (the y-values) that the function can produce.
When dealing with the function compositions like

• \( f \circ g \)
• \( g \circ f \)

we compute the domain based on where both component functions are defined.
For example, in the composition \( f^{\circ} g = (1 - \sqrt{x})^2 \), since the function \( g(x) \) must be defined, \( x \) should be non-negative, giving a domain of \([0, \infty)\). Additionally, the range resulted from the values the composition can reach, such as being all non-negative values that start from zero and go towards infinity.
Similarly, for \( g^{\circ} f = 1 - |x| \), the domain is all real numbers because it starts with a function \( f(x) \) defined for all real numbers. The range here reaches from \(-\infty\) to 1 because \( 1 - |x| \) can never be more than 1, introducing transformations from the composition functions.

Square root functions, like \( g(x) = 1 - \sqrt{x} \), have specific properties that influence both domain and range.

• The domain of a square root function includes all values for which the expression inside the square root is non-negative, since square roots of negative numbers are not defined in the set of real numbers.
• This means for \( g(x) = 1 - \sqrt{x} \), \( x \) must be greater than or equal to zero. As a result, the domain is \([0, \infty)\).

When calculating compositions involving square roots, each step must maintain non-negative constraints. For example, within the composition \( f \circ g \), which evaluates to \( (1 - \sqrt{x})^2 \), this domain restriction persists but affects the output in its range by transforming values to vary from zero upwards as \( x \) increases, due to the squaring effect neutralizing potential negative impacts.

The absolute value function, represented here as \( |x| \), is used in the composition \( g(f(x)) = 1 - |x| \).

• The absolute value of a number returns its distance from zero on the number line, essentially removing any negative sign, so it's always non-negative.
• This property is valuable in ensuring outputs that rely on symmetry around zero, such as when calculating expressions like \( \sqrt{x^2} = |x| \).

Through the composition \( g \circ f \), one sees that \( |x| \) maintains function integrity over all real numbers, allowing a domain of \((-\infty, \infty)\). The range, on the other hand, adjusts to \((-\infty, 1]\) because subtracting non-negative values (\(|x|\)) from 1 provides outputs that peak at 1 and extend downwards without bound. The understanding of absolute value functions reinforces how compositions can invert or maintain expected value outputs.