Find the domain of each of the following functions (i) f(x)=√(|x|-2) (ii) f(x)=√(4-|x|) (iii) f(

step 1 Our goal for this problem is to find the domain. So we're going to start off with F of X equals

Find the domain of each of the following functions (i)...

Find the domain of each of the following functions (i) $f(x)=\sqrt{|x|-2}$ (ii) $f(x)=\sqrt{4-|x|}$ (iii) $f(x)=\sqrt{|x|-1}+\sqrt{4-\lfloor x \mid}$ (iv) $f(x)=\sqrt{\frac{|x|-1}{3-|x|}}$ (v) $f(x)=\sqrt{|x|-x}$ (vi) $f(x)=\frac{1}{\sqrt{|x|-x}}$ (vii) $f(x)=\frac{1}{\sqrt{x-|x|}}$ viii) $f(x)=\sqrt{[x]-x}$ (ix) $f(x)=\frac{1}{\sqrt{[x]-x}}$ (x) $f(x)=\frac{1}{\sqrt{x-[x]}}$

Find the domain of each of the following functions
(i) $f(x)=\sqrt{|x|-2}$
(ii) $f(x)=\sqrt{4-|x|}$
(iii) $f(x)=\sqrt{|x|-1}+\sqrt{4-\lfloor x \mid}$
(iv) $f(x)=\sqrt{\frac{|x|-1}{3-|x|}}$
(v) $f(x)=\sqrt{|x|-x}$
(vi) $f(x)=\frac{1}{\sqrt{|x|-x}}$
(vii) $f(x)=\frac{1}{\sqrt{x-|x|}}$
viii) $f(x)=\sqrt{[x]-x}$
(ix) $f(x)=\frac{1}{\sqrt{[x]-x}}$
(x) $f(x)=\frac{1}{\sqrt{x-[x]}}$

Key Concepts

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. Determining the domain involves ensuring that all operations within the function, such as square roots and divisions, are valid for the given inputs. This concept is central to understanding where a function will work without resulting in undefined expressions like division by zero or square roots of negative numbers.

Square Root Function Restrictions

For functions involving square roots, it is essential to ensure that the expression inside the square root (the radicand) is nonnegative, because the square root of a negative number is not defined in the set of real numbers. This requirement leads to solving inequalities to identify the domain where the radicand is greater than or equal to zero.

Absolute Value Properties

The absolute value function returns the distance of a number from zero, making all outputs nonnegative. This property is useful when working with expressions that need to remain nonnegative, and it often simplifies or alters the conditions for domain restrictions when included in a function argument.

Floor Function (Greatest Integer Function)

The floor function, denoted by [x], returns the greatest integer less than or equal to x. This function is characterized by its discontinuous, piecewise constant nature, and understanding it is key when it appears in expressions whose validity depends on integer boundaries or when solving inequalities involving the floor function.

Rational Expression Constraints

When a function includes a rational expression (a fraction with a variable expression in the denominator), it is crucial to ensure that the denominator is never zero, as division by zero is undefined. This adds an extra step when determining the domain, requiring the identification and exclusion of values that cause the denominator to vanish.

Inequality Solving Techniques

Many functions involve components that impose inequality constraints on the variable, such as the nonnegativity of a radicand or the non-zero condition for a denominator. Mastery of solving these inequalities, often through case analysis and understanding of piecewise definitions, is critical for accurately determining the domain of a function.

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