Given $f(x)=3 x+1, g(x)=-2 x^2, h(x)=\frac{1}{3} \sqrt{x}-1$, determine the value of $\frac{1}{16} f(-1)-h(81)+g(2)$
Answer (1)
Evaluate f ( − 1 ) : f ( − 1 ) = 3 ( − 1 ) + 1 = − 2 .
Evaluate h ( 81 ) : h ( 81 ) = 3 1 81 − 1 = 2 .
Evaluate g ( 2 ) : g ( 2 ) = − 2 ( 2 ) 2 = − 8 .
Substitute the values into the expression: 16 1 ( − 2 ) − 2 + ( − 8 ) = − 10.125 . The final answer is − 10.125 .
Explanation
Understanding the Problem We are given three functions: f ( x ) = 3 x + 1 , g ( x ) = − 2 x 2 , and h ( x ) = 3 1 x − 1 . Our goal is to find the value of the expression 16 1 f ( − 1 ) − h ( 81 ) + g ( 2 ) .
Evaluating f(-1) First, we need to evaluate f ( − 1 ) . Substituting x = − 1 into the function f ( x ) , we get: f ( − 1 ) = 3 ( − 1 ) + 1 = − 3 + 1 = − 2
Evaluating h(81) Next, we evaluate h ( 81 ) . Substituting x = 81 into the function h ( x ) , we get: h ( 81 ) = 3 1 81 − 1 = 3 1 ( 9 ) − 1 = 3 − 1 = 2
Evaluating g(2) Now, we evaluate g ( 2 ) . Substituting x = 2 into the function g ( x ) , we get: g ( 2 ) = − 2 ( 2 ) 2 = − 2 ( 4 ) = − 8
Substituting and Simplifying Now we substitute the values we found into the expression 16 1 f ( − 1 ) − h ( 81 ) + g ( 2 ) :
16 1 f ( − 1 ) − h ( 81 ) + g ( 2 ) = 16 1 ( − 2 ) − 2 + ( − 8 ) = − 16 2 − 2 − 8 = − 8 1 − 10 = − 0.125 − 10 = − 10.125
Final Answer Therefore, the value of the expression is − 10.125 .
Examples
Understanding functions and evaluating expressions is crucial in many real-world applications. For example, in physics, you might use functions to model the trajectory of a projectile or the growth of a population. In finance, functions can represent the value of an investment over time. By evaluating these functions at specific points, you can make predictions and informed decisions. In this case, we evaluated three different functions and combined their results to find a final value, which is a common task in many scientific and engineering fields.
