91Ó°ÊÓ

When dealing with functions like the inverse sine function, understanding "domain and range" is crucial. The domain refers to the set of all possible input values (x-values) for which the function is defined. For a function like \(f(x)=\sin^{-1}(g(x))\), the domain is determined by the possible values of \(g(x)\) that fall within the standard domain of the inverse sine function, which is \([-1, 1]\).

In our given function \(f(x)=\sin^{-1}(|x-1|-2)\), the expression \(|x-1|-2\) must also meet this condition. Thus, to find the domain, we need to ensure that:

• \(-1 \leq |x-1|-2 \leq 1\)

Solving these inequalities will provide the allowable x-values. Solving \(|x-1|-2\geq -1\), gives us \(|x-1|\geq 1\). And solving \(|x-1|-2\leq 1\), results in \(|x-1|\leq 3\). This tells us that the domain of \(f(x)\) in our context is \{ 3 \leq x \leq 5 \} (initially inferred from step-by-step analysis). The corresponding range of the output values (f(x)) when \(f(x)\) does exist would be within the standard range of inverse sine \([-\pi/2, \pi/2]\).

Absolute value functions, denoted as \(|x|\), are essential in understanding how changes and shifts in input values influence the output. The absolute value function takes any real number and maps it to its non-negative part. For instance, both 2 and -2 mapped via \(|x|\) will result in 2.

In the context of the function \(f(x)=\sin^{-1}(|x-1|-2)\), \(|x-1|\) first computes the difference between \(x\) and 1, and then makes it non-negative. This means, it has a crucial role in cases like when \(x < 1\), turning what would be negative into positive values.

The piece \(|x-1|-2\) points out that there is a subtraction of 2 from the absolute value. So, you get downward shifts in the graph and outputs based on the threshold set by the values within the inverse sine domain limits. Understanding this shift helps map possible outputs accurately.

Piecewise functions allow us to define functions whose definition changes depending on segments or parts of the domain. Instead of being defined by a single expression, a piecewise function can have multiple expressions, each corresponding to a specific interval of the input variable.

Our original function \(f(x)=\sin^{-1}(|x-1|-2)\) is implicitly piecewise over different segments of \(x\):

• For \(x < 1\), the function is undefined since \(|x-1|-2\) results in values outside the domain of the inverse sine function.
• At \(x = 1\), \(|x-1|-2\) still takes a value outside of \([-1, 1]\), yielding it undefined.
• For \(1 < x < 3\), the function is still undefined as the inputs to \(\sin^{-1}\) don’t fit within its domain.
• For \(x = 3\), \(|x-1|-2 = 0\), which is in the domain of \(\sin^{-1}\), giving \(f(3)=0\).
• For \(x > 3\), but \(x \leq 5\), \(|x-1|-2\) stays within \([0, 1]\), hence defines proper outputs.

A piecewise approach facilitates understanding how specific ranges of \(x\) affect the function's real-world applicability, informing of its behavior under different conditions.