$\begin{array}{l}
h(x)=x^2+1 \quad k(x)=x-2 \\
(h+k)(2)=\square
\end{array}$
Evaluate $3 h(2)+2 k(3)=$ $\square$
Answer (1)
First, find h ( 2 ) by substituting x = 2 into h ( x ) = x 2 + 1 , which gives h ( 2 ) = 5 .
Next, find k ( 3 ) by substituting x = 3 into k ( x ) = x − 2 , which gives k ( 3 ) = 1 .
Then, substitute these values into the expression 3 h ( 2 ) + 2 k ( 3 ) .
Finally, calculate 3 ( 5 ) + 2 ( 1 ) = 15 + 2 = 17 , so the answer is 17 .
Explanation
Problem Analysis We are given two functions, h ( x ) = x 2 + 1 and k ( x ) = x − 2 . We need to evaluate the expression 3 h ( 2 ) + 2 k ( 3 ) . This involves substituting the given values into the functions and then performing the arithmetic operations.
Calculate h(2) First, let's find the value of h ( 2 ) . We substitute x = 2 into the function h ( x ) : h ( 2 ) = ( 2 ) 2 + 1 = 4 + 1 = 5
Calculate k(3) Next, let's find the value of k ( 3 ) . We substitute x = 3 into the function k ( x ) : k ( 3 ) = ( 3 ) − 2 = 1
Evaluate the Expression Now, we substitute the values of h ( 2 ) and k ( 3 ) into the expression 3 h ( 2 ) + 2 k ( 3 ) : 3 h ( 2 ) + 2 k ( 3 ) = 3 ( 5 ) + 2 ( 1 ) = 15 + 2 = 17
Final Answer Therefore, the value of the expression 3 h ( 2 ) + 2 k ( 3 ) is 17.
Examples
Understanding function evaluation is crucial in many real-world applications. For instance, in physics, if h ( t ) represents the height of an object at time t , and k ( v ) represents the air resistance at velocity v , then 3 h ( 2 ) + 2 k ( 3 ) could represent a combined effect of the object's height at t = 2 seconds and the air resistance at v = 3 m/s, scaled by certain factors. Evaluating such expressions helps in modeling and predicting physical phenomena.
