If [tex]T(x)=\sqrt[3]{28 x-76}[/tex], evaluate:
a) [tex]T(5)=[/tex] $\square$
b) [tex]T(-5)=[/tex] $\square$
Answer (1)
Substitute x = 5 into T ( x ) and find T ( 5 ) .
Calculate 28 ( 5 ) − 76 = 140 − 76 = 64 .
Evaluate 3 64 = 4 .
Substitute x = − 5 into T ( x ) and find T ( − 5 ) .
Calculate 28 ( − 5 ) − 76 = − 140 − 76 = − 216 .
Evaluate 3 − 216 = − 6 .
The final answers are 4 and − 6 .
Explanation
Understanding the Problem We are given the function T ( x ) = 3 28 x − 76 . We need to evaluate T ( 5 ) and T ( − 5 ) .
Evaluating T(5) First, let's evaluate T ( 5 ) . We substitute x = 5 into the expression for T ( x ) : T ( 5 ) = 3 28 ( 5 ) − 76 T ( 5 ) = 3 140 − 76 T ( 5 ) = 3 64 T ( 5 ) = 4
Evaluating T(-5) Next, let's evaluate T ( − 5 ) . We substitute x = − 5 into the expression for T ( x ) :
T ( − 5 ) = 3 28 ( − 5 ) − 76 T ( − 5 ) = 3 − 140 − 76 T ( − 5 ) = 3 − 216 T ( − 5 ) = − 6
Final Answer Therefore, T ( 5 ) = 4 and T ( − 5 ) = − 6 .
Examples
Imagine you are designing a temperature control system where the required cooling or heating ( T ) depends on an input value ( x ) representing the current ambient temperature. The function T ( x ) = 3 28 x − 76 models this relationship. Evaluating T ( 5 ) and T ( − 5 ) helps you determine the system's output at specific ambient temperatures, allowing you to calibrate the system for optimal performance. This kind of function evaluation is crucial in engineering for designing and testing control systems.
