An equation that defines y as a function of x is given. Find ƒ(3). y+2x²=3-x

Verified step by step guidance

1. Start with the given equation: \(y + 2x^{2} = 3 - x\).
2. Isolate \(y\) on one side to express it as a function of \(x\): subtract \$2x^{2}\( from both sides to get \)y = 3 - x - 2x^{2}$.
3. Rewrite the function notation as \(f(x) = 3 - x - 2x^{2}\) to clearly show \(y\) as a function of \(x\).
4. To find \(f(3)\), substitute \(x = 3\) into the function: \(f(3) = 3 - (3) - 2(3)^{2}\).
5. Simplify the expression step-by-step by calculating the powers and performing the arithmetic to find the value of \(f(3)\).

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to exactly one output y. Evaluating ƒ(3) means substituting x = 3 into the function's equation and calculating the corresponding y-value.

Solving for y in Terms of x

To find ƒ(3), the equation must be rewritten to express y explicitly as a function of x. This involves isolating y on one side of the equation, often by performing algebraic operations like addition, subtraction, or division.

Substitution Method

After expressing y as a function of x, substitution involves replacing x with the given value (here, 3) to compute the specific output. This step is essential to find the function's value at a particular input.