If $f(x)=2 x^2+1$, what is $f(x)$ when $x=3$ ?
The objective is to evaluate the function \(f(x) = 2x^2 + 1\) at \(x = 3\). A function represents a rule where each input is mapped to a specific output. In this case, for any input \(x\), we square it, multiply the result by 2, and then add 1.
To find \(f(3)\), we replace the variable \(x\) in the function expression with the value 3:
\[f(3) = 2(3)^2 + 1\]
Following the order of operations, we first evaluate the exponent:
\[3^2 = 9\]
Next, we perform the multiplication:
\[2 \times 9 = 18\]
Finally, we add 1:
\[18 + 1 = 19\]
Thus, the value of the function at \(x=3\) is 19.
Evaluating functions is a fundamental skill in many fields, including economics and physics. For example, if a company's production cost is modeled by the function \(C(x) = 2x^2 + 1\), where \(x\) represents the number of batches produced, calculating \(C(3)\) allows the company to determine the exact cost of producing 3 batches. By applying this algebraic method, we can make informed decisions about resource allocation and budget planning.