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Function evaluation is a process where we find the value of a function for specific values of its variable, in this case, the variable is \(x\). In the given problem, the function is \(f(x) = \frac{3x + 5}{2}\). By substituting values \(x = 0\), \(x = 1\), and \(x = -1\) into the function, we can calculate the corresponding outputs.

• For \(x = 0\), substitute 0 into the equation: \(f(0) = \frac{3(0) + 5}{2} = \frac{5}{2}\).
• For \(x = 1\), substitute 1: \(f(1) = \frac{3(1) + 5}{2} = \frac{8}{2} = 4\).
• For \(x = -1\), substitute -1: \(f(-1) = \frac{3(-1) + 5}{2} = \frac{2}{2} = 1\).

Function evaluation helps us understand how the function behaves at different points. It is fundamental in analyzing functions and predicting outcomes.

Linear functions are mathematical functions that create a straight line when graphed. They can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The function in this problem \(f(x) = \frac{3x + 5}{2}\) is indeed a linear function.

Let's simplify it to see its linear form more clearly. We can rewrite the function as:
\[ f(x) = \frac{3}{2}x + \frac{5}{2} \]
Here, \(\frac{3}{2}\) is the slope and \(\frac{5}{2}\) is the y-intercept. This slope tells us how steep the line is and in which direction it moves. In our function, a positive slope means the line rises as \(x\) increases. Linear functions are very common in mathematical modeling due to their straightforwardness.

Solving equations involves finding the values of the variable that make the equation true. In this exercise, we solve for \(x\) when \(f(x)\) is equal to given values: 0, 1, and -1.

• For \(f(x) = 0\): Write \(0 = \frac{3x + 5}{2}\). Solving for \(x\), we get \(x = -\frac{5}{3}\).
• For \(f(x) = 1\): Use \(1 = \frac{3x + 5}{2}\). Solving gives \(x = -\frac{3}{2}\).
• For \(f(x) = -1\): Set \(-1 = \frac{3x + 5}{2}\). Solving this results in \(x = -\frac{9}{2}\).

Through solving these equations, we identify specific \(x\) values that satisfy each condition, thus enhancing our understanding of how the function behaves at these points.

Mathematical problem solving involves using processes and strategies to find solutions to mathematical questions. In the context of this exercise, it requires understanding the problem, planning a solution strategy, executing the calculations, and reviewing the results.

To address this exercise:

• Understand the function and what is required by reading the problem statement thoroughly.
• Evaluate and compare the function at given \(x\) values.
• Solve for \(x\) where the function equals specific values.
• Review the outcomes to ensure they are logical and satisfy the initial conditions.

Good problem solvers go through these stages systematically, adapting their approach as needed. This structured methodical method not only improves accuracy but also fosters deeper mathematical understanding.