- Identify the correct piece of the function by comparing the input \(x = -5\) to the conditions: since \(-5 < 1\), the applicable expression is \(f(x) = 2x - 1\).
- Substitute \(x = -5\) into the expression: \(f(-5) = 2(-5) - 1\).
- Calculate the result: \(f(-5) = -10 - 1 = -11\).
- The final value is \(\boxed{-11}\).
Explanation
- Identifying the Correct Piece of the Function
We are given a piecewise-defined function \(f(x)\) which takes different expressions depending on the value of \(x\):
\[ f(x) = \begin{cases} -3x + 1 & \text{if } x > 1 \ -2x & \text{if } x = 1 \ 2x - 1 & \text{if } x < 1 \end{cases} \]
To determine the value of \(f(-5)\), we must evaluate the condition of the input \(x = -5\). Comparing \(x = -5\) to the given conditions, we see that \(-5 < 1\). Therefore, we must use the part of the function definition corresponding to \(x < 1\), which is \(f(x) = 2x - 1\).
- Evaluating the Function at x = -5
Now that we have identified the appropriate expression, we substitute \(x = -5\) into the function \(f(x) = 2x - 1\):
\[ f(-5) = 2(-5) - 1 \]
Performing the multiplication:
\[ 2(-5) = -10 \]
Subtracting 1 from this result:
\[ -10 - 1 = -11 \]
Thus, the value of the function at \(x = -5\) is \(-11\). Comparing this to the given options, the correct choice is \(-11\).
Examples
Piecewise functions are used in various fields, such as finance for calculating tax brackets where different tax rates apply based on income level, or in engineering for controlling systems that behave differently under different temperatures or pressures. For instance, a heating control system might turn on the heater if the temperature is below a certain threshold, keep it off if it is within a comfortable range, and start a cooling fan if it exceeds another, higher threshold, all of which can be modeled by a piecewise function.