Consider a function f(x) satisfying the condition f(2 + x) = 2x^3 + 3; then |f(4)| is.
Understand the Problem
The question is asking us to find the value of the function f at x=4, given a specific condition of the function that involves determining its behavior for other values.
Answer
$|f(4)| = 19$
Answer for screen readers
The value of ( |f(4)| ) is ( 19 ).
Steps to Solve
- Understand the function's condition
We are given the condition for the function ( f ): $$ f(2 + x) = 2x^3 + 3 $$
- Substitute for ( x )
To find ( f(4) ), we need to express 4 in terms of ( 2 + x ): $$ 4 = 2 + x \implies x = 4 - 2 = 2 $$
- Use the condition to find ( f(4) )
Substitute ( x = 2 ) into the condition: $$ f(4) = f(2 + 2) = 2(2)^3 + 3 $$
Calculate ( 2(2)^3 + 3 ): $$ 2(2^3) = 2 \cdot 8 = 16 $$ Then add 3: $$ 16 + 3 = 19 $$
- Calculate the absolute value
The problem asks for ( |f(4)| ): $$ |f(4)| = |19| = 19 $$
The value of ( |f(4)| ) is ( 19 ).
More Information
This problem involves manipulating functions based on given conditions. The absolute value is taken to ensure the answer is non-negative, which is relevant in many real-world scenarios.
Tips
- Forgetting to convert ( 4 ) to ( 2 + x ) correctly.
- Miscalculating the expression ( 2x^3 + 3 ).
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