Given f(x)=√(x) and g(x)=|x-3|, find (g)/(f) . Determine the domain of the function in interval

step 1 Okay, the first thing we know here is that the square root is defined only for positive numbers

Given f(x)=√(x) and g(x)=|x-3|, find (g)/(f) . Determine the...

Key Concepts

Absolute Value Function

The absolute value function, represented by |x|, measures the distance of a number from zero on the real number line, regardless of its sign. It is defined for all real numbers because every real number has a non-negative absolute value. Understanding this property helps when combining the absolute value function with other functions, as it does not usually impose additional restrictions on the domain.

Determining the Domain of a Function

Determining the domain of a function involves identifying all input values for which the function is defined. When dealing with operations such as division, it is necessary to consider the individual domains of the separate functions and then exclude any values that lead to undefined operations, such as division by zero. The final domain is often expressed in interval notation to clearly indicate the ranges of valid inputs.

Division of Functions

Division of functions involves creating a new function by dividing the output of one function by the output of another. It is essential to determine where both functions are defined and to ensure that the divisor is never zero, as division by zero is undefined. This process requires finding the intersection of the domains of the individual functions and excluding any values that cause the denominator to be zero.

Square Root Function

The square root function, often denoted as ?x, is defined for non-negative values of x because the square root of a negative number is not a real number. Hence, the domain of the square root function is the set of all x such that x is greater than or equal to zero. This inherent restriction on the input values is crucial when the square root function is part of a composite function.

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