$f(x)=x$
$g(x)=1$
What is the domain of $\left(\frac{g}{f}\right)(x) ?$
A. $x \neq 0$
B. $x \neq-1$
C. All real numbers
Answer (1)
Find the expression for the combined function: ( f g ) ( x ) = x 1 .
Identify values where the function is undefined: x = 0 .
Exclude these values from the set of all real numbers.
Express the domain in interval notation: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . The domain is all real numbers except 0, so the answer is x = 0 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x and g ( x ) = 1 . We want to find the domain of the function ( f g ) ( x ) .
Finding the Expression First, we need to find the expression for ( f g ) ( x ) . This is simply the function g ( x ) divided by the function f ( x ) : ( f g ) ( x ) = f ( x ) g ( x ) = x 1 .
Determining the Domain Now, we need to determine the domain of the function x 1 . The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is undefined when the denominator is equal to zero. So, we need to find the values of x for which x = 0 .
Expressing the Domain The function x 1 is undefined when x = 0 . Therefore, the domain of the function is all real numbers except for x = 0 . In interval notation, this is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if you're modeling the speed of a car as a function of time, you need to know the domain to ensure that the time values you're using are valid. Similarly, in physics, when dealing with quantities like electric potential which can be inversely proportional to distance, understanding the domain helps avoid division by zero, which is physically meaningless. This concept is also vital in economics when dealing with cost functions or revenue functions, where certain input values might not be feasible or make sense.
