The functions $f$ and $g$ are defined as follows.
[tex]
\begin{array}{l}
f(x)=\frac{x^2}{x-1} \\
g(x)=\frac{x-3}{x^2-7 x+12}
\end{array}
[/tex]
For each function, find the domain.
Write each answer as an interval or union of intervals.
Answer (1)
To find the domain of f ( x ) = x − 1 x 2 , set the denominator x − 1 = 0 , which gives x = 1 . The domain is ( − ∞ , 1 ) ∪ ( 1 , ∞ ) .
To find the domain of g ( x ) = x 2 − 7 x + 12 x − 3 , factor the denominator as ( x − 3 ) ( x − 4 ) . Set ( x − 3 ) ( x − 4 ) = 0 , which gives x = 3 and x = 4 . The domain is ( − ∞ , 3 ) ∪ ( 3 , 4 ) ∪ ( 4 , ∞ ) .
The domain of f ( x ) is ( − ∞ , 1 ) ∪ ( 1 , ∞ ) .
The domain of g ( x ) is ( − ∞ , 3 ) ∪ ( 3 , 4 ) ∪ ( 4 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x − 1 x 2 and g ( x ) = x 2 − 7 x + 12 x − 3 . 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 − 1 x 2 , the denominator is x − 1 . We need to find the values of x for which x − 1 = 0 . Solving for x , we get x = 1 . Therefore, the domain of f ( x ) is all real numbers except x = 1 . In interval notation, this is ( − ∞ , 1 ) ∪ ( 1 , ∞ ) .
Finding the Domain of g(x) For the function g ( x ) = x 2 − 7 x + 12 x − 3 , the denominator is x 2 − 7 x + 12 . We need to find the values of x for which x 2 − 7 x + 12 = 0 . We can factor the quadratic expression as ( x − 3 ) ( x − 4 ) . So, we have ( x − 3 ) ( x − 4 ) = 0 . This means x = 3 and x = 4 . Therefore, the domain of g ( x ) is all real numbers except x = 3 and x = 4 . In interval notation, this is ( − ∞ , 3 ) ∪ ( 3 , 4 ) ∪ ( 4 , ∞ ) .
Final Answer Therefore, the domain of f ( x ) is ( − ∞ , 1 ) ∪ ( 1 , ∞ ) , and the domain of g ( x ) is ( − ∞ , 3 ) ∪ ( 3 , 4 ) ∪ ( 4 , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if we are modeling the population growth of a species, the domain of the function would represent the time interval over which the model is valid. Similarly, in physics, if we are modeling the trajectory of a projectile, the domain would represent the time interval during which the projectile is in motion. In economics, the domain of a cost function might represent the range of production levels for which the function is defined. In all these cases, understanding the domain helps us to interpret the results of the model and make meaningful predictions.
