Solving and Verifying a Function with Algebraic Operations

Given the function ( f(x) -2x - 3^2 cdot 4 ), if ( 2f(5 cdot 7) ) is plugged in, what would be the answer? The textbook key says -1, but many initially struggle to obtain that answer. This article will walk through the correct steps to arrive at the solution and help clarify the process.

Step-by-Step Solution

Step 1: Calculate ( f(5) )

Substitute ( x 5 ) into the function:

( f(5) -2(5) - 3^2 cdot 4 )

Calculate ( 5 - 3^2 ):

( 5 - 3^2 5 - 9 -4 )

Now substitute back into the function:

( f(5) -2(-4) cdot 4 )

Carry out the multiplication:

( f(5) 8 cdot 4 -4 )

Step 2: Calculate ( 2f(5 cdot 7) )

Substitute ( f(5) ) into the expression:

( 2f(5 cdot 7) 2(-4) cdot 7 )

Perform the multiplication:

( 2(-4) cdot 7 -8 cdot 7 -56 )

Is this the correct process? Let's verify the question and the steps:

Revised Approach

Given the function ( f(x) -2x - 3^2 cdot 4 ), if we evaluate ( f(5) ) and denote the value as ( y ), then ( 2y cdot 7 -1 ).

Let's correct the steps:

( f(5) -2(5 - 3^2) cdot 4 )

( f(5) -2(5 - 9) cdot 4 )

( f(5) -2(-4) cdot 4 )

( f(5) 8 cdot 4 )

( f(5) -4 )

( 2f(5) cdot 7 2(-4) cdot 7 )

( 2f(5) cdot 7 -8 cdot 7 )

( 2f(5) cdot 7 -56 )

Clarification and Verification

Let's ensure that we are correctly evaluating the function and the expression:

( f(5) -2(5 - 3^2) cdot 4 )

( f(5) -2(5 - 9) cdot 4 )

( f(5) -2(-4) cdot 4 )

( f(5) 8 cdot 4 )

( f(5) -4 )

( 2f(5 cdot 7) 2(-4) cdot 7 )

( 2f(5 cdot 7) -8 cdot 7 )

( 2f(5 cdot 7) -56 )

It appears that the textbook key might be assuming a different function or a different interpretation of the expression. Therefore, we need to ensure the function and the expression are correctly evaluated.

Final Answer

Thus, the correct solution for ( 2f(5 cdot 7) ) is:

boxed{-56}

Conclusion

The process of evaluating functions and expressions in algebra is crucial in understanding and solving mathematical problems. By carefully following the order of operations and correctly substituting values, we can solve complex equations and verify our results.

Additional Tips for Solving Similar Problems

1. **Identify and Substitute Correctly**: Always substitute the given value into the function correctly and follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

2. **Double-Check Your Work**: Make sure to re-evaluate your steps to ensure no mistakes were made in arithmetic or in the order of operations.

3. **Understand the Function**: Before evaluating the function, ensure you fully understand the function itself. Sometimes, the way the function is presented can lead to confusion. Make sure everything is clearly defined and correctly interpreted.

4. **Use of Resources**: If you are still unsure, refer to textbooks, online resources, or consult with a teacher or tutor to gain clarity.

By following these steps and tips, you can solve similar problems with greater confidence and accuracy.