In which function is $x=2$ mapped to 32?
We are given four different functions: \(f(x) = -3x^2 - 4\), \(g(x) = 4(x + 3)^2 - 68\), \(h(x) = 3x\), and \(j(x) = 2x - 62\). We need to determine which of these functions maps the input \(x = 2\) to the output value of \(32\). To do this, we will systematically substitute \(x = 2\) into each function and calculate the result.
Substitute \(x = 2\) into each function:
\(f(2) = -3(2)^2 - 4 = -3(4) - 4 = -12 - 4 = -16\).
\(g(2) = 4(2 + 3)^2 - 68 = 4(5)^2 - 68 = 4(25) - 68 = 100 - 68 = 32\).
\(h(2) = 3(2) = 6\).
\(j(2) = 2(2) - 62 = 4 - 62 = -58\).
Comparing the calculated results with the target value of \(32\), we see that \(g(2) = 32\). Therefore, the function \(g(x) = 4(x + 3)^2 - 68\) is the correct function. The final answer is \(\boxed{g(x)=4(x+3)^2-68}\).
Functions are essential in programming and engineering to process inputs and produce specific outputs. For example, in a game, a function might calculate the damage dealt to an enemy based on the player's weapon level, or in a financial model, a function might calculate compound interest based on the time deposited. By substituting specific values into these models, we can predict outcomes accurately, just as we identified which mathematical formula satisfied the given target condition.