The functions $f$ and $g$ are defined as follows.
[tex]
\begin{array}{l}
f(x)=\frac{x+7}{x^2-49} \\
g(x)=\frac{x}{x^2+81}
\end{array}
[/tex]
For each function, find the domain.
Write each answer as an interval or union of intervals.
Domain of $f$ : $\square$
Domain of $g$ : $\square$
Answer (1)
Find the domain of f ( x ) by setting the denominator x 2 − 49 not equal to zero, which gives x = 7 and x = − 7 .
Express the domain of f ( x ) in interval notation: ( − ∞ , − 7 ) ∪ ( − 7 , 7 ) ∪ ( 7 , ∞ ) .
Find the domain of g ( x ) by setting the denominator x 2 + 81 not equal to zero, which has no real solutions.
Express the domain of g ( x ) in interval notation: ( − ∞ , ∞ ) .
Explanation
Understanding the Problem We are given two functions: f ( x ) = x 2 − 49 x + 7 and g ( x ) = x 2 + 81 x . We need to find the domain of each function. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.
Finding the Domain of f(x) For the function f ( x ) = x 2 − 49 x + 7 , we need to find the values of x for which the denominator x 2 − 49 is equal to zero. So, we solve the equation x 2 − 49 = 0 .
Expressing the Domain of f(x) in Interval Notation We can factor x 2 − 49 as ( x − 7 ) ( x + 7 ) = 0 . Thus, x = 7 or x = − 7 . Therefore, the domain of f ( x ) is all real numbers except x = 7 and x = − 7 . In interval notation, this is ( − ∞ , − 7 ) ∪ ( − 7 , 7 ) ∪ ( 7 , ∞ ) .
Finding the Domain of g(x) For the function g ( x ) = x 2 + 81 x , we need to find the values of x for which the denominator x 2 + 81 is equal to zero. So, we solve the equation x 2 + 81 = 0 .
Expressing the Domain of g(x) in Interval Notation This gives x 2 = − 81 . Since x is a real number, there is no real solution for x . Therefore, the denominator x 2 + 81 is never zero for any real number x . Thus, the domain of g ( x ) is all real numbers. In interval notation, this is ( − ∞ , ∞ ) .
Final Answer Therefore, the domain of f ( x ) is ( − ∞ , − 7 ) ∪ ( − 7 , 7 ) ∪ ( 7 , ∞ ) and the domain of g ( x ) is ( − ∞ , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, if f ( x ) represents the cost of producing x items, the domain tells us the possible number of items we can produce. Similarly, if g ( x ) represents the population size at time x , the domain tells us the valid time intervals for which the model is applicable. In engineering, knowing the domain helps in determining the range of input values for which a system operates safely and effectively. For example, the domain of a function describing the stress on a bridge might exclude values that would cause the bridge to collapse.
