A function is defined by $f(x)=2 x^2-3$. What is the value of $f(-2)$ ?
The problem asks us to evaluate the quadratic function \(f(x) = 2x^2 - 3\) for a specific input value \(x = -2\). A function is a rule that assigns a unique output for every input in its domain. In this case, we have a quadratic function where the input \(x\) is first squared, multiplied by 2, and then 3 is subtracted from the result. To find \(f(-2)\), we need to substitute \(-2\) for every instance of \(x\) in the expression.
Now we substitute \(x = -2\) into the expression \(f(x) = 2x^2 - 3\):
\[f(-2) = 2(-2)^2 - 3.\]
First, we calculate the square of \(-2\):
\[(-2)^2 = (-2) \times (-2) = 4.\]
Next, we multiply this result by the coefficient 2:
\[2 \times 4 = 8.\]
Finally, we subtract 3 from the result:
\[8 - 3 = 5.\]
Thus, the value of the function at \(x = -2\) is \(5\). Comparing this to the given choices A: -19, B: -11, C: 5, and D: 13, we see that the correct choice is C.
Functions are widely used in physics and engineering to model real-world phenomena. For example, if you are calculating the position of an object moving under constant acceleration, such as a ball falling from a height, you use a quadratic function like \(d(t) = \frac{1}{2}gt^2 + v_0t + d_0\). By evaluating this function at specific times, you can determine exactly where the object will be, which is crucial for tasks like designing safe structures or predicting the trajectory of a projectile.