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

Verified step by step guidance

1. Start with the given equation: \(y + 2x^{2} = 3 - x\).
2. Rewrite the equation to express \(y\) explicitly as a function of \(x\): \(y = 3 - x - 2x^{2}\).
3. To find \(ƒ(3)\), substitute \(x = 3\) into the function: \(y = 3 - (3) - 2(3)^{2}\).
4. Simplify the expression step-by-step: calculate \$3\(, then subtract \)3\(, and then subtract \)2\( times \)3$ squared.
5. Evaluate the powers and multiplications carefully, then combine all terms to find the value of \(ƒ(3)\).

Key Concepts

Function Definition and Notation

A function defines a relationship where each input x corresponds to exactly one output y, often written as y = f(x). Understanding function notation, such as f(3), means evaluating the function at x = 3 to find the corresponding y-value.

Solving for y in Terms of x

To find f(3), you must first express y explicitly as a function of x by isolating y in the given equation. This involves algebraic manipulation to rewrite the equation in the form y = f(x), enabling direct substitution of x-values.

Substitution and Evaluation

Once y is expressed as f(x), substitute the given value x = 3 into the function. Then, perform the necessary arithmetic operations to compute the output f(3), which gives the value of y corresponding to x = 3.