Find the domain of each of the following functions (i) f(x)=√(x-2) (ii) f(x)=√(x+5) (iii) f(x)=√

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) $f(x)=\sqrt{x+5}$ (iii) $f(x)=\sqrt{4-x}$ (iv) $f(x)=\sqrt{x-2}+\sqrt{4-x}$ (v) $f(x)=\sqrt{x-3}-\sqrt{7-x}$ (vi) $f(x)=\sqrt{\frac{x-2}{5-x}}$ (vii) $f(x)=\sqrt{\frac{x}{x-1}}$ (viii) $f(x)=\sqrt{1-\frac{1}{x}}$ (ix) $f(x)=\sqrt{\frac{1}{x}-1}$ (x) $f(x)=\sqrt{\frac{x-2}{x}}$

Find the domain of each of the following functions
(i) $f(x)=\sqrt{x-2}$
(ii) $f(x)=\sqrt{x+5}$
(iii) $f(x)=\sqrt{4-x}$
(iv) $f(x)=\sqrt{x-2}+\sqrt{4-x}$
(v) $f(x)=\sqrt{x-3}-\sqrt{7-x}$
(vi) $f(x)=\sqrt{\frac{x-2}{5-x}}$
(vii) $f(x)=\sqrt{\frac{x}{x-1}}$
(viii) $f(x)=\sqrt{1-\frac{1}{x}}$
(ix) $f(x)=\sqrt{\frac{1}{x}-1}$
(x) $f(x)=\sqrt{\frac{x-2}{x}}$

Key Concepts

Intersection of Domain Conditions

Many functions include multiple components, each with its own domain restrictions. The overall domain of the function is found by taking the intersection of these restrictions, ensuring that all conditions are simultaneously satisfied for a given input value.

Domain Restrictions in Rational Functions

When a function includes a rational expression, it is crucial to ensure that the denominator is never zero, as division by zero is undefined. This additional restriction must be considered alongside any other domain constraints, such as those imposed by square roots.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. It is fundamental to understanding any function, as it specifies the allowable values that can be substituted into the function's equation without leading to undefined expressions or non-real results.

Square Root Function Restrictions

For functions involving square roots, the expression inside the square root (known as the radicand) must be greater than or equal to zero to yield a real number output. This requirement sets up an inequality involving the radicand that must be solved to determine the appropriate domain.

Inequality Solving for Domain

Determining the domain often involves solving inequalities, particularly when the function contains square roots or any expressions that are constrained by the need to remain nonnegative. Correctly solving these inequalities is key to identifying which input values produce valid outputs.

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