If [tex]T(x)=\sqrt{5 x+4}[/tex], evaluate:
a) [tex]T(9)=[/tex] $\square$
b) [tex]T(-4)=[/tex] $\square$
Note: Use integers or reduced fractions. If the resulting value is not real, enter "DNE."
Answer (1)
Substitute x = 9 into T ( x ) = 5 x + 4 and evaluate: T ( 9 ) = 5 ( 9 ) + 4 = 49 = 7 .
Substitute x = − 4 into T ( x ) = 5 x + 4 and evaluate: T ( − 4 ) = 5 ( − 4 ) + 4 = − 16 , which is not a real number.
Therefore, T ( 9 ) = 7 and T ( − 4 ) is DNE.
The final answers are 7 and D NE .
Explanation
Understanding the Problem We are given the function T ( x ) = 5 x + 4 , and we need to evaluate T ( 9 ) and T ( − 4 ) . This involves substituting the given values into the function and simplifying. If the value inside the square root is negative, the result is not a real number, and we denote it as 'DNE' (Does Not Exist).
Calculating T(9) To find T ( 9 ) , we substitute x = 9 into the function: T ( 9 ) = 5 ( 9 ) + 4 T ( 9 ) = 45 + 4 T ( 9 ) = 49 T ( 9 ) = 7
Calculating T(-4) To find T ( − 4 ) , we substitute x = − 4 into the function: T ( − 4 ) = 5 ( − 4 ) + 4 T ( − 4 ) = − 20 + 4 T ( − 4 ) = − 16 Since we have a negative number inside the square root, the result is not a real number. Therefore, T ( − 4 ) does not exist.
Final Answer Therefore, T ( 9 ) = 7 and T ( − 4 ) does not exist (DNE).
Examples
Imagine you are designing a bridge and need to calculate the tension on a cable. The tension might be modeled by a function similar to T ( x ) = 5 x + 4 , where x represents some external load. Evaluating T ( 9 ) would tell you the tension when the load is 9 units, and finding that T ( − 4 ) is not real might indicate a critical condition where the model is no longer valid or the design is flawed under certain negative loads. This helps engineers ensure the bridge's safety and stability under various conditions.
