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Function evaluation is a fundamental concept in algebra, where you substitute a specific value for the variable in the function's formula to calculate the output. Think of a function as a little machine: you feed it an input, and it spits out an output according to a predetermined rule.

For instance, consider the function in the exercise, given as \( h(x) = 6x + 1 \). To evaluate this function for \( x = \frac{1}{2} \), you replace every occurrence of \( x \) in the function's formula with \( \frac{1}{2} \). The steps go as follows:

1. Replace \( x \) with \( \frac{1}{2} \) in the formula, resulting in \( h(\frac{1}{2}) = 6 \times \frac{1}{2} + 1 \).
2. Simplify the expression by multiplying 6 with \( \frac{1}{2} \), which gives 3, and add 1 to get the final answer.
3. Therefore, \( h(\frac{1}{2}) \) equals 4.

Mastering function evaluation is important, as it is used frequently in solving equations, graphing functions, and understanding how changes in input affect the output.

Linear functions are the simplest form of functions where the relationship between the input, denoted as x, and the output, denoted as y or f(x), can be represented as a straight line on the coordinate plane. The general form of a linear function is \( y = mx + b \), where:\

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The function \( h(x) = 6x + 1 \) from the exercise is a linear function because it can be rewritten in the form \( y = 6x + 1 \), where the slope \( m \) is 6, and the y-intercept \( b \) is 1. No matter what value you plug in for \( x \), the corresponding \( y \) or \( h(x) \) value is determined by the same rule, making the graph of the function a straight line.

The slope of a line represents how steep the line is and the direction it slopes. To visualize this, think about hiking uphill or downhill: the steeper the hill, the greater the slope. The slope is calculated as the change in y (vertical change) over the change in x (horizontal change), often represented as \( m = \frac{\text{rise}}{\text{run}} \).

A positive slope means the line is increasing, tilting uphill from left to right. A negative slope means the line is decreasing, tilting downhill from left to right. If the slope is zero, the line is horizontal, indicating no change in y for changes in x. Conversely, if the line is vertical, the slope is undefined.

In the context of our exercise, the function \( h(x) = 6x + 1 \) has a slope of 6 which is positive, indicating that the line is increasing as x increases. This positive, constant slope is why the function is considered one-to-one since each input maps to a unique output, constructing a consistently upward moving line with no repeating values.