- Identify the correct piece of the function for each input based on the domain conditions: \(f(-1) = 9(-1)+4\), \(f(0) = \sqrt{9(0)^2+16}\), and \(f(2) = \sqrt{9(2)^2+16}\).
- Evaluate \(f(-1) = -5\).
- Evaluate \(f(0) = 4\).
- Evaluate \(f(2) = \sqrt{52} \approx 7.21\).
- Final answers: \(f(-1) = \boxed{-5}\), \(f(0) = \boxed{4}\), \(f(2) \approx \boxed{7.21}\).
Explanation
- Analysis of the Piecewise Function Definitions
The given piecewise function is defined as:
\[ f(x)=\left\{\begin{array}{cc} 9 x+4 & x<0 \ \sqrt{9 x^2+16} & x \geq 0 \end{array}\right. \]
To evaluate the function at specific points, we must first determine which domain interval each point belongs to.
- For \(x = -1\), the value is less than 0, so we use the first piece: \(f(x) = 9x+4\).
- For \(x = 0\), the value satisfies \(x \geq 0\), so we use the second piece: \(f(x) = \sqrt{9x^2+16}\).
- For \(x = 2\), the value is greater than 0, so we use the second piece: \(f(x) = \sqrt{9x^2+16}\).
- Evaluating f(-1)
To calculate \(f(-1)\), we use the condition \(x < 0\):
\[f(-1) = 9(-1) + 4 = -9 + 4 = -5\]
The result of this calculation is \(-5\).
- Evaluating f(0)
To calculate \(f(0)\), we use the condition \(x \geq 0\):
\[f(0) = \sqrt{9(0)^2 + 16} = \sqrt{0 + 16} = \sqrt{16} = 4\]
The result of this calculation is \(4\).
- Evaluating f(2)
To calculate \(f(2)\), we use the condition \(x \geq 0\):
\[f(2) = \sqrt{9(2)^2 + 16} = \sqrt{9(4) + 16} = \sqrt{36 + 16} = \sqrt{52}\]
Calculating the square root, \(\sqrt{52} \approx 7.2111\). Rounding to two decimal places, we get approximately \(7.21\).
- Final Results Summary
In summary, the values for the given function are:
\(f(-1) = -5\)
\(f(0) = 4\)
\(f(2) \approx 7.21\)
Examples
Piecewise functions are commonly used in engineering to model scenarios that change behavior depending on certain thresholds, such as tax brackets or electricity pricing. For example, a power utility might charge a lower rate for the first 100 kWh of energy consumed and a higher rate for every kWh beyond that threshold, which can be expressed mathematically as a piecewise function. By calculating these values, engineers can accurately estimate costs or predict system performance under different conditions.