Problem 671

For the pair of function f and g, determine the domain of f/g.

F(x) = \frac{2}{x-3},g(x) = 4-x




The domain of t/g is the set of all values common to the domains of t and g, excluding values for which g(x) is 0.

The domain of a function defined by an equation is the set of all number for which real values of the function can be calculated. A function can be undefined at a number because calculating its value results in an undefined in an undefined operation like division by zero or an even root of a negative number.

For the function , f(x) =\frac{2}{x-3}. The denominator is zero for x= 3, which result in an undefined value. Therefore, the domain of f is {x I x is  a real number and x\neq3}.

For the function , g(x) = 4 – x, a value can be calculated for any real number x. Therefore, the domain of g is {x I x is a real number}.

The domain of the quotient also excludes all values for which f(x) is zero. Therefore solve the equation

4 – x = 0.

X = 4

The domain of f/g is the set of all values common to { x I x is a real number and x \neq3} and {x I x is a real number}, and also excluding x = 4.

Therefore , the domain of f/g is {x I x is a real number and x \neq 3 and  x \neq4}.


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