If f(x) = 2x - x 2, then what is the value of f(x + 2) + f(x - 2) when x = 0?
-8
Understanding the Function f(x)
The problem provides a function defined as \(f(x) = 2x - x^2\). This is a polynomial function, specifically a quadratic function because the highest power of \(x\) is 2. We need to evaluate an expression involving this function at a specific value of \(x\).
Calculating f(x+2)
To find \(f(x+2)\), we substitute \((x+2)\) wherever we see \(x\) in the original function definition \(f(x) = 2x - x^2\).
The expression becomes:
\(f(x+2) = 2(x+2) - (x+2)^2\)
Now, we expand and simplify the expression:
\(f(x+2) = 2x + 4 - (x^2 + 2(x)(2) + 2^2)\)
\(f(x+2) = 2x + 4 - (x^2 + 4x + 4)\)
\(f(x+2) = 2x + 4 - x^2 - 4x - 4\)
Combine like terms:
\(f(x+2) = -x^2 + (2x - 4x) + (4 - 4)\)
\(f(x+2) = -x^2 - 2x\)
Calculating f(x-2)
Similarly, to find \(f(x-2)\), we substitute \((x-2)\) wherever we see \(x\) in the original function definition \(f(x) = 2x - x^2\).
The expression becomes:
\(f(x-2) = 2(x-2) - (x-2)^2\)
Now, we expand and simplify the expression:
\(f(x-2) = 2x - 4 - (x^2 - 2(x)(2) + 2^2)\)
\(f(x-2) = 2x - 4 - (x^2 - 4x + 4)\)
\(f(x-2) = 2x - 4 - x^2 + 4x - 4\)
Combine like terms:
\(f(x-2) = -x^2 + (2x + 4x) + (-4 - 4)\)
\(f(x-2) = -x^2 + 6x - 8\)
Finding the Sum f(x+2) + f(x-2)
Now we add the expressions we found for \(f(x+2)\) and \(f(x-2)\):
\(f(x+2) + f(x-2) = (-x^2 - 2x) + (-x^2 + 6x - 8)\)
\(f(x+2) + f(x-2) = -x^2 - 2x - x^2 + 6x - 8\)
Combine like terms:
\(f(x+2) + f(x-2) = (-x^2 - x^2) + (-2x + 6x) - 8\)
\(f(x+2) + f(x-2) = -2x^2 + 4x - 8\)
So, the expression \(f(x+2) + f(x-2)\) simplifies to \(-2x^2 + 4x - 8\).
Evaluating the Expression at x = 0
We need to find the value of \(f(x+2) + f(x-2)\) when \(x = 0\). We can substitute \(x = 0\) into the simplified expression \(-2x^2 + 4x - 8\).
Substitute \(x=0\):
\(f(0+2) + f(0-2) = -2(0)^2 + 4(0) - 8\)
Perform the calculations:
\(= -2(0) + 0 - 8\)
\(= 0 + 0 - 8\)
\(= -8\)
Final Result for f(x+2) + f(x-2) at x = 0
The value of \(f(x+2) + f(x-2)\) when \(x = 0\) is \(-8\).
Step-by-Step Evaluation at x=0 (Alternative Method)
Alternatively, we could first evaluate \(f(x+2)\) and \(f(x-2)\) at \(x=0\) separately and then add the results.
When \(x=0\):
- \(x+2\) becomes \(0+2 = 2\)
- \(x-2\) becomes \(0-2 = -2\)
So we need to find \(f(2) + f(-2)\).
Using the original function \(f(x) = 2x - x^2\):
First, calculate \(f(2)\):
\(f(2) = 2(2) - (2)^2\)
\(f(2) = 4 - 4\)
\(f(2) = 0\)
Next, calculate \(f(-2)\):
\(f(-2) = 2(-2) - (-2)^2\)
\(f(-2) = -4 - (4)\)
\(f(-2) = -4 - 4\)
\(f(-2) = -8\)
Finally, add \(f(2)\) and \(f(-2)\):
\(f(2) + f(-2) = 0 + (-8)\)
\(f(2) + f(-2) = -8\)
Both methods yield the same result, confirming that the value of \(f(x+2) + f(x-2)\) when \(x = 0\) is \(-8\).
Revision Table: Key Concepts in Functions
| Concept | Description | Example (using f(x) = 2x - x<sup>2</sup>) |
|---|---|---|
| Function Notation | \(f(x)\) represents the output of the function for a given input \(x\). | \(f(3) = 2(3) - 3^2 = 6 - 9 = -3\) |
| Evaluating a Function | Substituting a specific value or expression for the variable (e.g., \(x\)). | To evaluate \(f(a)\), replace every \(x\) with \(a\). |
| Function Transformation (Substitution) | Replacing \(x\) with an expression like \((x+c)\) or \((x-c)\) changes the function's graph and the resulting expression. | \(f(x+2)\) means replacing \(x\) with \((x+2)\) in the formula for \(f(x)\). |
Additional Information: Polynomial Functions
The function \(f(x) = 2x - x^2\) is a type of polynomial function. Polynomials are expressions consisting of variables and coefficients, that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- A quadratic function is a polynomial function of degree 2, meaning the highest exponent is 2. Its general form is \(ax^2 + bx + c\), where \(a \ne 0\).
- In our case, \(f(x) = -x^2 + 2x\), so \(a=-1\), \(b=2\), and \(c=0\).
- Evaluating polynomial functions involves substituting the input value into the expression and performing the arithmetic operations.
- Polynomials are fundamental in algebra and calculus and are used to model various phenomena.
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