- Use the condition \(x < 1\) to evaluate \(f(-2) = -(-2)^2 + 7 = 3\).
- Use the condition \(x \geq 1\) to evaluate \(f(1) = -2(1)^2 + 7 = 5\).
- Use the condition \(x \geq 1\) to evaluate \(f(2) = -2(2)^2 + 7 = -1\).
- Final results: \(f(-2) = \boxed{3}\), \(f(1) = \boxed{5}\), \(f(2) = \boxed{-1}\).
Explanation
- Analysis of the Piecewise Function
To determine the values of the function \(f(x)\), we must observe the definition provided:
\[f(x)=\begin{cases} -x^2+7 & x<1 \ -2x^2+7 & x \geq 1 \end{cases}\]
The function is defined piecewise, meaning we choose which expression to use based on the input value \(x\). If the input \(x\) is less than 1, we use the expression \(-x^2 + 7\). If the input \(x\) is greater than or equal to 1, we use the expression \(-2x^2 + 7\).
- Calculating \(f(-2)\)
For \(x = -2\), we compare the input to the condition \(x < 1\). Since \(-2 < 1\), we apply the first expression:
\[f(-2) = -(-2)^2 + 7\]
First, square \(-2\) to get \(4\), then apply the negative sign to get \(-4\). Adding \(7\):
\[f(-2) = -4 + 7 = 3.\]
- Calculating \(f(1)\)
For \(x = 1\), we compare the input to the condition \(x \geq 1\). Since \(1 \geq 1\), we apply the second expression:
\[f(1) = -2(1)^2 + 7\]
First, square \(1\) to get \(1\), then multiply by \(-2\) to get \(-2\). Adding \(7\):
\[f(1) = -2 + 7 = 5.\]
- Calculating \(f(2)\)
For \(x = 2\), we compare the input to the condition \(x \geq 1\). Since \(2 \geq 1\), we apply the second expression:
\[f(2) = -2(2)^2 + 7\]
First, square \(2\) to get \(4\), then multiply by \(-2\) to get \(-8\). Adding \(7\):
\[f(2) = -8 + 7 = -1.\]
Examples
Piecewise functions are used in many real-world scenarios, such as calculating income tax. For instance, the tax rate might be 10% for the first \$\(20,000 earned and 20% for any income exceeding that amount. In this case, the tax function would be defined in two parts, mirroring how we split the domain here to find the correct output based on the input. Understanding how to handle different conditions allows for precise modeling of tiered systems, whether it is finance, energy usage pricing, or even tiered shipping costs based on weight.