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Linear functions are straightforward and predictable. They represent a straight line when graphed. A basic linear function is written as: \( f(x) = mx + b \) where \(m\) is the slope, and \(b\) is the y-intercept. The slope indicates how steep the line is, while the y-intercept is where the line crosses the y-axis. In the exercise, the linear function given is: \(f(x) = -3x + 4\). Here, the slope (\(m\)) is -3, indicating a downward motion, and the y-intercept (\(b\)) is 4. You simply replace \(x\) with any given value to evaluate it. For example, \(f(4)\) means substituting 4 for \(x\) and calculating: \(f(4) = -3(4) + 4 = -12 + 4 = -8\). This calculation tells us that the value of \(f\) at \(x = 4\) is -8.

Quadratic functions add complexity because they form parabolas when graphed. They have the general form: \(g(x) = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. If \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards. In the exercise, the quadratic function is: \(g(x) = -x^2 + 4x + 1\). Here, \(a = -1\), \(b = 4\), and \(c = 1\), meaning the parabola opens downward. To evaluate \(g(4)\), substitute 4 for \(x\): \(g(4) = - (4)^2 + 4(4) + 1 = -16 + 16 + 1 = 1\). Thus, the value of \(g\) at \(x = 4\) is 1. Quadratic functions often require more steps to evaluate due to the \(x^2\) term.

Function operations involve combining functions through addition, subtraction, multiplication, or division. They help create new functions from existing ones. The basic operations are:

• **Addition**: \((f + g)(x) = f(x) + g(x)\)
• **Subtraction**: \((f - g)(x) = f(x) - g(x)\)
• **Multiplication**: \((f \times g)(x) = f(x) \times g(x)\)
• **Division**: \((f / g)(x) = f(x) / g(x)\) (provided \(g(x) eq 0\))

In our exercise, we use the subtraction operation. We first evaluate \(f(4)\) and \(g(4)\), then subtract \(g(4)\) from \(f(4)\): $$ f(4) - g(4) = -8 - 1 = -9 $$. This shows the result of \(f - g\) at \(x = 4\) is -9. Understanding function operations is crucial for tackling complex problems.