If $R(x)=\sqrt{4 x-4}$, evaluate:
a) $R(17)=$ $\square$
b) $R(-13)=$ $\square$
Note: Use integers or reduced fractions. If the resulting value is not real, enter "DNE."
Answer (1)
Substitute x = 17 into R ( x ) = 4 x − 4 and simplify: R ( 17 ) = 4 ( 17 ) − 4 = 64 = 8 .
Substitute x = − 13 into R ( x ) = 4 x − 4 and simplify: R ( − 13 ) = 4 ( − 13 ) − 4 = − 56 .
Since the value inside the square root is negative, R ( − 13 ) is not a real number.
The final answers are R ( 17 ) = 8 and R ( − 13 ) = DNE .
Explanation
Understanding the Problem We are given the function R ( x ) = 4 x − 4 and asked to evaluate R ( 17 ) and R ( − 13 ) . We need to substitute the given values into the function and simplify. If the value inside the square root is negative, the result is not a real number, so we will write 'DNE'.
Evaluating R(17) First, let's evaluate R ( 17 ) . We substitute x = 17 into the expression for R ( x ) : R ( 17 ) = 4 ( 17 ) − 4 R ( 17 ) = 68 − 4 R ( 17 ) = 64 R ( 17 ) = 8
Evaluating R(-13) Next, let's evaluate R ( − 13 ) . We substitute x = − 13 into the expression for R ( x ) : R ( − 13 ) = 4 ( − 13 ) − 4 R ( − 13 ) = − 52 − 4 R ( − 13 ) = − 56 Since the value inside the square root is negative, R ( − 13 ) is not a real number.
Conclusion Therefore, R ( − 13 ) = DNE .
Examples
Understanding functions like R ( x ) = 4 x − 4 is crucial in various real-world applications. For instance, in physics, this could represent the velocity of an object under certain conditions, where x is a variable like time or distance. Knowing how to evaluate such functions allows us to predict the object's velocity at different points. Similarly, in engineering, these functions can model stress or strain on materials, helping engineers design safer and more efficient structures. The ability to substitute values into functions and interpret the results is a fundamental skill in many scientific and technical fields.
