Verified Answer :
- For \(f(x) = 2x - 1\) and domain \(D = \{-4, -2, 0, 2, 4\}\), the evaluated values are \(\boxed{\{-9, -5, -1, 3, 7\}}\).
- For \(f(t) = -3(t - 2)\) and domain \(D = \{-4, -2, 0, 2, 4\}\), the evaluated values are \(\boxed{\{18, 12, 6, 0, -6\}}\).
Explanation
- Problem Analysis and Understanding the Domain
We are tasked with evaluating two linear functions, \(f(x) = 2x - 1\) and \(f(t) = -3(t - 2)\), for a specific set of input values provided by the domain \(D = \{-4, -2, 0, 2, 4\}\). Evaluating a function means substituting each value from the domain into the variable of the function and calculating the resulting value. This process helps us understand how the output of these functions changes in relation to their inputs.
- Evaluating \(f(x) = 2x - 1\)
To evaluate \(f(x) = 2x - 1\) for \(x \in \{-4, -2, 0, 2, 4\}\), we substitute each value into the expression:
- For \(x = -4\): \(f(-4) = 2(-4) - 1 = -8 - 1 = -9\)
- For \(x = -2\): \(f(-2) = 2(-2) - 1 = -4 - 1 = -5\)
- For \(x = 0\): \(f(0) = 2(0) - 1 = 0 - 1 = -1\)
- For \(x = 2\): \(f(2) = 2(2) - 1 = 4 - 1 = 3\)
- For \(x = 4\): \(f(4) = 2(4) - 1 = 8 - 1 = 7\)
Thus, the values for \(f(x)\) are \(\{-9, -5, -1, 3, 7\}\).
- Evaluating \(f(t) = -3(t - 2)\)
To evaluate \(f(t) = -3(t - 2)\) for \(t \in \{-4, -2, 0, 2, 4\}\), we substitute each value into the expression:
- For \(t = -4\): \(f(-4) = -3(-4 - 2) = -3(-6) = 18\)
- For \(t = -2\): \(f(-2) = -3(-2 - 2) = -3(-4) = 12\)
- For \(t = 0\): \(f(0) = -3(0 - 2) = -3(-2) = 6\)
- For \(t = 2\): \(f(2) = -3(2 - 2) = -3(0) = 0\)
- For \(t = 4\): \(f(4) = -3(4 - 2) = -3(2) = -6\)
Thus, the values for \(f(t)\) are \(\{18, 12, 6, 0, -6\}\).
Examples
Functions are fundamental in modeling real-world situations, such as calculating taxi fares or predicting expenses. For instance, if a taxi charges a base fee and a rate per mile, we can write a function for the total cost \(C(m) = \text{base} + \text{rate} \times m\). By substituting different mileages into this function, you can quickly determine the cost for various trip lengths, helping you plan your budget effectively. Similarly, evaluating functions for a specific domain allows you to see how outputs change systematically as inputs increase or decrease.