The function is defined by . What is the value of ?
For questions asking for a value like when a function is defined by a formula, immediately rewrite the formula with replaced by the given number in parentheses, then simplify using the order of operations: exponents first, then multiplication, then addition/subtraction. Be especially careful with negative inputs and squared terms—always write the negative number in parentheses before squaring to avoid sign mistakes, and double-check each arithmetic step to avoid simple calculation errors.
Hints
Meaning of
Think about what it means to evaluate a function at a specific value. How do you use the formula to find ?
Substitute carefully
Rewrite the formula, putting everywhere you see . Use parentheses around when you substitute into and .
Order of operations
After substituting, do the exponent first, then the multiplications, and finally add all the terms together. Pay close attention to the negative signs in and in .
Check each arithmetic step
Once you have three numbers (from , , and ), add them carefully. If your total does not come from adding three positive numbers, recheck your signs.
Desmos Guide
Evaluate in Desmos
In Desmos, type the expression 4(-2)^2 - 3(-2) + 5 exactly as written (with parentheses around -2). The value that Desmos outputs for this expression is the value of .
Step-by-step Explanation
Understand what means
The function is defined by . The expression means: take the formula for and replace every with , then simplify.
Substitute into the function
Write the expression for by substituting for :
Be sure to include parentheses around so the square applies to the entire negative number.
Evaluate powers and multiplication
First, calculate the square:
Now multiply:
So the expression becomes:
Add the terms to get the final value
Add the numbers step by step:
So , which corresponds to answer choice D.