Asked by Julian Torres on Jun 10, 2024
Verified
Find the domain of the rational function. h(t) =10t2t2−4t−45h ( t ) = \frac { 10 t ^ { 2 } } { t ^ { 2 } - 4 t - 45 }h(t) =t2−4t−4510t2
A) (−∞,0) ∪(0,∞) ( - \infty , 0 ) \cup ( 0 , \infty ) (−∞,0) ∪(0,∞)
B) (−∞,∞) ( - \infty , \infty ) (−∞,∞)
C) (−∞,−5) ∪(−5,0) ∪(0,∞) ( - \infty , - 5 ) \cup ( - 5,0 ) \cup ( 0 , \infty ) (−∞,−5) ∪(−5,0) ∪(0,∞)
D) (−∞,0) ∪(0,9) ∪(9,∞) ( - \infty , 0 ) \cup ( 0,9 ) \cup ( 9 , \infty ) (−∞,0) ∪(0,9) ∪(9,∞)
E) (−∞,−5) ∪(−5,9) ∪(9,∞) ( - \infty , - 5 ) \cup ( - 5,9 ) \cup ( 9 , \infty ) (−∞,−5) ∪(−5,9) ∪(9,∞)
Domain
The set of all possible input values for a function.
Rational Function
A function represented by the ratio of two polynomials.
- Pinpoint and delineate the domain of rational functions.
Verified Answer
MA
Michael AlemuJun 15, 2024
Final Answer :
E
Explanation :
The domain of a rational function is all real numbers except where the denominator equals zero. To find these points for h(t)=10t2t2−4t−45h(t) = \frac{10t^2}{t^2 - 4t - 45}h(t)=t2−4t−4510t2 , set the denominator equal to zero and solve for ttt : t2−4t−45=0t^2 - 4t - 45 = 0t2−4t−45=0 . Factoring the quadratic equation gives (t−9)(t+5)=0(t - 9)(t + 5) = 0(t−9)(t+5)=0 , so t=9t = 9t=9 and t=−5t = -5t=−5 are the points where the function is undefined. Thus, the domain is all real numbers except t=9t = 9t=9 and t=−5t = -5t=−5 , which corresponds to (−∞,−5)∪(−5,9)∪(9,∞)( - \infty , - 5 ) \cup ( - 5,9 ) \cup ( 9 , \infty )(−∞,−5)∪(−5,9)∪(9,∞) .
Learning Objectives
- Pinpoint and delineate the domain of rational functions.