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Linear functions are mathematical expressions that create a straight line when plotted on a graph. They typically come in the form:

• \( f(x) = ax + b \)

Here, \( a \) and \( b \) are constants. The value \( a \) is called the slope, and it determines the steepness of the line, while \( b \) represents the y-intercept, which is where the line crosses the y-axis.
For the given function \( f(x) = -3x + 1 \), \( a = -3 \) and \( b = 1 \). The slope is -3, meaning the line goes downward as you move from left to right. The y-intercept is 1, so the line crosses the y-axis at the point (0, 1).
To evaluate the function at a specific value of \( x \), you replace \( x \) with that value. For example, in the problem, we substituted \( x \) with 2:

• \( f(2) = -3(2) + 1 = -6 + 1 = -5 \)

This means that at \( x = 2 \), the output \( f(2) \) is -5.

Quadratic functions are equations that can be written in the form:

• \( g(x) = ax^2 + bx + c \)

These functions plot as a parabola (a U-shaped curve) on a graph. The constants \( a \), \( b \), and \( c \) determine the shape and position of the parabola:

• The value \( a \) affects whether the parabola opens upwards or downwards.
• If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• The values \( b \) and \( c \) affect the width and position.

In our problem, the quadratic function is \( g(x) = x^2 + 2 \). Here, \( a = 1 \) (the parabola opens upwards), \( b = 0 \), and \( c = 2 \).
To evaluate this function at a given \( x \), substitute the value into the function. For instance, evaluating at \( x = 2 \) involves:

• \( g(2) = (2)^2 + 2 = 4 + 2 = 6 \)

This means that at \( x = 2 \), the output \( g(2) \) is 6.

The substitution method is a fundamental mathematical technique used to simplify and solve equations. It's pivotal in evaluating functions. Here's how the method works step-by-step:

• Identify the function you need to evaluate, such as \( f(x) \) or \( g(x) \).
• Substitute the given value of \( x \) into the function.
• Simplify the resulting expression to find the output.

In our problem, we needed to find \( f(2) + g(2) \):

• First, we substituted 2 into \( f(x) \) to get \( f(2) = -3(2) + 1 = -5 \).
• Next, we substituted 2 into \( g(x) \) to get \( g(2) = (2)^2 + 2 = 6 \).
• Finally, we added these results together: \( f(2) + g(2) = -5 + 6 = 1 \).

The substitution method simplifies complex expressions by breaking them into manageable steps, making it easier to understand and solve equations.